This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
0 with a constant C."/, the latter constant may or may not remain bounded as " ! 0. In fact, in R3 (as well as in all odd dimensions) there is a Laurent expansion near z D 0 of the form .H
z2/
1
DB
2
z
2
CB
1
z
1
C B0 C B1 z C : : : C Bk z k C Rk .z/ (3.76)
where kw Rk .z/w k2!2 Ck jzjkC1 provided is large enough depending on k. B 2 is the orthogonal projection onto the zero-energy eigenspace. Zero energy being regular, or equivalently (3.75), is the same as B 2 D B 1 D 0. The relevance of this condition for dispersive estimates can be seen from the fact that various evolutions of H appear as some form of Fourier transform of the spectral measure. More specifically, e
itH
1
Z Pc .H / D
e it E.d/Pc .H /
0
e
˙it
p
H C1
Z Pc .H / D
1
e
˙it
p
(3.77) C1
E.d/Pc .H /
0
where E./ is the spectral resolution of H and Pc is the projection onto the continuous spectral subspace of H . The first line in (3.77) is the Schrödinger evolution, whereas the second line represents Klein–Gordon evolutions associated with H . The connection with the resolvent is furnished by the well-known identity between projection-valued measures on L2 E.d/Pac .H / D Im.H
/
1
Pac .H / d;
>0
(3.78)
with Pac .H / being the projection onto the absolutely continuous subspace of H . The right-hand side of (3.78) is singular at D 0 if B 1 ¤ 0. In view of (3.77), it is therefore no surprise and also well-known that for the case where zero energy
3
This is called “limiting absorption principle”.
119
3.4 Dispersive estimates for the perturbed linear evolution
is singular, the usual dispersive estimates fail4 (in both the pointwise and Strichartz senses). There is a sizable body of literature on dispersive estimates for both wave and Schrödinger evolutions with regular threshold, the main objective being to establish the same decay properties as for the free evolutions. However, we cannot possibly review this literature here (see however [123], [124]). If the potential V exhibits sufficient decay and regularity, then Yajima [145] found a general and powerful approach based on the wave operators. Yajima’s theory allows one to deduce the desired dispersive bounds for H D H0 C V directly from those for the free equation (this applies to both pointwise as well as Strichartz estimates, as well as to different kinds of evolutions).
3.4.1 The wave operators in scattering theory The wave operators are defined as the strong limits in L2 W WD s
lim e
t !1
itH itH0
e
;
H WD
C V;
H0 D
(3.79)
which means that Wf D lim t!1 e itH e itH0 f for any f 2 L2 . It is easy to see that this limit exists in R3 provided that V 2 L2 .R3 /. Indeed, let f be a Schwartz function, say, and note that5 Z t Z t d isH isH0 i tH i tH0 e e f ds D i e isH Ve isH0 f ds : e e f f D 0 0 ds 3
Using the unitarity of the e isH evolution, as well the hsi 2 pointwise decay of the free evolution, one concludes that the integral on the right is absolutely convergent in L2 over .0; 1/ provided V 2 L2 . Finally, the Schwartz class is dense in L2 which implies that the limit exists for all f 2 L2 due to the unitarity of the evolution operators. The wave operators are isometries on L2 . Thus W W and W W are projections, the former onto Ran.W /, the latter being the identity (this is a general property of isometries; consider the right shift in `2 over the positive integers as an example). 4
5
In this regard, recall Howland’s razor [75], which states that resonances cannot be defined in terms of a single operator on an abstract Hilbert space. However, one can detect them by means of a single operator on Lp with p ¤ 2. This argument is called “Cook’s method”.
120
3 Above the ground state energy I: Near Q
By construction one has the intertwining property e isH W D W e isH0 H) E./W D WE0 ./ where E; E0 refer to the spectral resolutions of H , and H0 , respectively. Therefore, Ran.W / L2a:c: , the absolutely continuous subspace of H . In fact, it is known that for “nice” potentials (such as V D 3Q2 ) this is an equality, and that there is no singular continuous spectrum, see Agmon [1], Kuroda [93], and Reed, Simon [120], Vol. 3, for example. Hence W W D Id;
W W D Pc .H / D Id
Ppp .H /
where Ppp is the projection onto the space spanned by all eigenfunctions. The latter space is finite dimensional for “nice” potentials, for example if the Birman– Schwinger operator 1
K WD jV j 2 . /
1
1
jV j 2
is a compact operator on L2 .R3 /. In particular, f .H /Pc .H / D Wf .H0 /W
(3.80)
for any bounded continuous f . This latter identity is the key for applying Yajima’s theorem to which we now turn.
3.4.2 Yajima’s Lp bound on the wave operators Yajima showed under certain conditions on V that W is bounded on Lp .R3 / for any 1 p 1. In R3 , the special case of his theorem from [145] which we used in the proof of Theorem 3.22 reads as follows. Needless to say, the following result applies to V D 3Q2 . Theorem 3.26. Let V be real-valued and jV .x/j . hxi , where > 5. Assume furthermore that zero energy is regular for H D C V . Then the wave operator W from (3.79) is bounded on Lp .R3 / for all 1 p 1. A typical example demonstrating the applicability of Yajima’s theorem already appeared in the proof of Theorem 3.22, see how (3.64) lead to (3.65). Some version of Theorem 3.26 holds in all dimensions, as well as for W k;p spaces.
3.4 Dispersive estimates for the perturbed linear evolution
121
3.4.3 Singular thresholds Generally speaking, this case is less understood. It is definitely “non-generic” and may therefore seem somewhat exotic. However, it does arise in actual problems, such as for the energy critical wave equation in R3 , see Chapter 6, Section 6.4. There the linearized operator associated with the ground state is LC D 5W 4 with 1 W .r/ D .1 C r 2 =3/ 2 , and LC exhibits a zero energy resonance but not an eigenvalue at zero energy – at least over the radial functions; nonradially, one has root modes rW which are eigenfunctions. In this situation it is not meaningful to seek general dispersive or Strichartz estimates6 since the threshold resonance destroys the free linear decay, at least for general data. However, this failure is restricted to a finite-dimensional subspace which we can hope to isolate (this is the statement that B 1 is of finite rank). The idea is that after subtracting the linear dynamics on this subspace we should be left with an evolution which does exhibit free dispersive estimates. As an example, consider the following result from [91]. In order to apply it, one specializes to H D 5W 4 restricted to radial functions. This is legitimate, since the potential is radial. Note that W 4 decays exactly like hxi 4 . Hence it is important that we do not require too much decay in Proposition 3.27. Proposition 3.27. Assume that V is a real-valued potential such that jV .x/j . hxi where > 3 is arbitrary but fixed. If H D C V has neither a resonance nor an eigenvalue at zero, then
sin.t pH /
(3.81) Pc f . t 1 kf kW 1;1 .R3 /
p 1 H for all t > 0. Now assume that zero is a resonance but not R an eigenvalue of H . Let be the unique resonance function normalized so that V .x/ dx D 1. Then there exists a constant c0 ¤ 0 such that
sin.t pH /
Pc f c0 . ˝ /f . t 1 kf kW 1;1 .R3 / (3.82)
p 1 H for all t > 0. Note that even the regular threshold assertion (3.81) does not follow from Yajima’s theorem since that result requires more decay, see Theorem 3.26. The ap6
Note, however, that Yajima did establish Lp boundedness of the wave operators even if zero energy is 0 where p D p .d /. singular, but in a restricted range 1 < p < p < p
122
3 Above the ground state energy I: Near Q
proach to the proof of Proposition 3.27 in [91] is based on the Laurent expansion (3.76) for small energies, and a Born expansion for large energies. Note that the pointwise decay in the above proposition is that of the free wave equation in three dimensions. We remark that Proposition 3.27 was used in [91] to construct Lipschitz graphs of codimension 1 near W with the property that solutions starting from this manifold exhibit the desired forward global existence and asymptotic stability property. However, the underlying topology in this construction was much stronger than the energy space and it remains an open problem to decide whether or not such a manifold also exists in the energy topology. See Section 6.4 for more on this issue.
3.4.4 Beceanu’s linear dispersive estimate In this section7 , we present some of the linear analysis introduced by Beceanu [8], [9] in his construction of the center-stable manifolds for the cubic NLS equation in the critical topology and without any symmetry assumptions. To be more specific, consider the nonlinear equation i@ t
D j j2 ;
which admits the periodic solitons e
it˛ 2
Q C ˛ 2 Q D Q3 ;
.t; x/ 2 R1C3
(3.83)
Q.x; ˛/. Here Q.x; ˛/ solves Q.x; ˛/ D ˛Q.˛x; 1/
where Q.x; 1/ > 0 is the ground state which we encountered in Chapter 2. It has been known since the work of Berestycki, Cazenave [11], Cazenave, Lions [28], and Weinstein [143], [144] that these periodic solutions are unstable; in fact, they can blow up under arbitrarily small perturbations. This is of course in analogy with the Klein–Gordon instability, cf. Corollary 3.1. An important difference between NLKG and NLS arises at the L2 -critical powers and below: while NLS solitons become stable for L2 -subcritical nonlinearities, this is not the case for NLKG; indeed, for the latter one has the Bates–Jones theorem on the existence of unstable manifolds which implies the instability of the ground states. In [122] it was shown, relative to a suitable topology, that NLS solitons (for the cubic nonlinearity) are nevertheless conditionally asymptotically stable; this of course means that asymptotic stability is guaranteed provided the data are chosen 7
Skip on first reading; this is only relevant to the NLS equation.
3.4 Dispersive estimates for the perturbed linear evolution
123
from a small Lipschitz hyper-surface containing the 8-dimensional soliton manifold. The drawback of this result lay with the choice of topology, which contained an L1 -component and which was therefore non-invariant. Beceanu [9] then carried out the construction in the energy topology (and in fact, the weaker critical topology of (3.83)), which yielded the center-stable manifold with the desired invariance property. His construction is based on a delicate Strichartz estimate for linear operators with small time-dependent, but space-independent, coefficients which we present in this section. A significant difference between the NLS and NLKG equations lies with the need to modulate the ground state for the former, even in the radial setting whereas for NLKG this is not the case and Q remains unchanged. This refers to the fact that 2 e i t ˛ Q is multiplied by an additional phase e i .t/ and ˛ becomes time-dependent, ˛ D ˛.t/; for nonradial solutions one also encounters space and momentum translations, adding another six parameters into the equation. We now linearize (3.83) around the ground state e it Q where Q D Q.; 1/, which leads to the following matrix operator (we view all operators in this section as complex linear ones): C 1 2Q2 .; 1/ Q2 .; 1/ HD ; Q2 .; 1/ 1 C 2Q2 .; 1/ see the following section for more details on this linearization. H is conjugate to 1h 1 2 i
1 i h1 H i 1
h 0 i i L i Di i LC 0
(3.84)
where LC D
C1
3Q.; 1/2 ;
L D
C1
Q.; 1/2 :
The equality (3.84) is to be considered as one between complex linear operators. However, it is also natural to view the left-hand side as acting on vectors uu12 with u1 ; u2 real-valued. In that case the right-hand side needs to be rewritten as hL
C
0
0 i D L: L
As already noted above, the spectral properties of LC ; L and especially H are quite delicate. First, we remark that L has the same gap property, see (3.12), as LC . For this we refer the reader to [44] and [41]. The following result summarizes what can be obtained by rigorous analysis, see [52], [76], [122], combined with some numerical findings, such as [44] and [99]. In the work of Marzuola, Simpson [99]
124
3 Above the ground state energy I: Near Q
numerics is used to assist index computations of certain quadratic forms in the spirit of the virial argument of Fibich, Merle, Raphael [54]. For simplicity, we restrict ourselves to the Hilbert space8 L2rad .R3 / in Proposition 3.28. Proposition 3.28. The essential spectrum of H is . 1; 1 [ Œ1; 1/ and there are no imbedded eigenvalues or resonances in the essential spectrum, the discrete spectrum is of the form f0; i; ig where > 0 with ˙i both simple eigenvalues, the root space at 0 is of dimension two, and the thresholds ˙1 are neither eigenvalues nor resonances. In explicit form, the root space is spanned by ! ! Q @˛ Q 0 D ; 0 D (3.85) Q @˛ Q and one has H0 D 0; H2 0 D 0. Let HG˙ D iG˙ with the normalization kG˙ k2 D 1. Then the eigenfunctions G˙ are exponentially decaying and of the form G˙ D gg˙ . ˙
Proof. The description of the root space of H goes back to [143], [144]. The imaginary spectrum was identified by Grillakis [63], [64] (see also [122]), and for the exponential decay of the corresponding eigenfunctions see [76]. All these results are based on purely analytical arguments. The fact that H does not have embedded eigenvalues in the essential spectrum was shown in [99], assisted by some numerical computations. Their proof also implies that there are no non–zero eigenvalues in the gap Œ 1; 1, and that the thresholds are not resonances. The latter two facts also follow rigorously by the analytical arguments in [122] combined with the proof of the gap properties of L˙ in [41]. Furthermore, it is to be expected that by combining the analytical argument in [99] with the rigorous study of the ground state conducted in [41] one can eliminate all numerical components from [99]. This would then establish the absence of embedded eigenvalues in the essential spectrum analytically, removing all numerical components from this proof. Next, we present a result for non-selfadjoint Schrödinger evolutions which originatesTin [8] (in fact, Beceanu proves a stronger result in Lorentz spaces). Let S D p;q Lpt .RC ; Lqx / be the Strichartz space with 2 p 1, 2 q 6, and 1 0 2 3 3 C q D 2 , and let S be its dual. For the remainder of this chapter, 3 D p 0 1 8
The only change to Proposition 3.28 is that H has a root space of dimension eight rather than two.
3.4 Dispersive estimates for the perturbed linear evolution
125
i
ess spec
1
0
ess spec
1
i
Figure 3.8. Spectrum of the linearized operator as in Proposition 3.28
is the third Pauli matrix. Also, we define the matrix operators H0 ; V via C1 0 H0 D ; H D H0 C V: 0 1 Lemma 3.29. Let A; a 2 L1 .R/ be real-valued and satisfy kAk1 C kak1 < c0 for some small absolute constant c0 . The solution 2 C RI L2 .R3 / \ C 1 RI H 2 .R3 / of the problem i @ t C H C iA.t/rPc C a.t/3 Pc D F 2 S ;
.0/ D 0 2 L2 .R3 / (3.86)
where Pc is the projection corresponding to the essential spectrum of H, obeys the Strichartz estimates kPc kS . k 0 kL2 .R3 / C kF kS :
(3.87)
Furthermore, if 0 2 H 1 , then krPc kS . k 0 kH 1 .R3 / C krF kS :
(3.88)
126
3 Above the ground state energy I: Near Q
Finally, one has scattering. Suppose for simplicity that A D 0. Then there exists 1 2 H 1 such that Pc .t/ D e i3
Rt
0.
C1Ca.s// ds
1 C o.1/
(3.89)
in H 1 as t ! 1. Proof. We follow [8], and set A D 0 for simplicity. Clearly, the proof should be perturbative in a by nature, with a D 0 being the nontrivial statement that Strichartz estimates (including the endpoint) hold for the equation i@ t C H D F : However, the latter has been established by several authors, see for example [8], Theorem 1.3 and [43]. Due to the lack of any physical localization of the a.t/ term, the perturbative analysis is nontrivial. On the other hand, note that any perturbation 6 of the form a.t/.x/ where the multiplier is bounded L6 .R3 / ! L 5 .R3 / can be taken to the right-hand side by virtue of the endpoint Strichartz estimate. To commence with the actual argument, consider the following auxiliary equation, with arbitrary but fixed ı > 0, and Pd D Id Pc : i@ t Z C HPc Z C iıPd Z C a.t/3 Pc Z D F
(3.90)
with data Z.0/ D 0 . We claim the Strichartz estimates for general data Z.0/, kZkS . kZ.0/kL2 .R3 / C kF kS :
(3.91)
Q If so, then ZQ WD Pc Z satisfies Z.0/ D Pc 0 and i@ t ZQ C HZQ C a.t/3 ZQ D Pc F C a.t/Œ3 ; Pc ZQ which is the same as the PRc projection of (3.86). Thus, Pc D ZQ and (3.91) imt plies (3.87). Let A.t/ D 0 a.s/ ds and write U.t/ D e iA.t/3 , Z.t/ D U.t/˚ . Then (3.90) becomes i@ t ˚ C U
1
.HPc C iıPd /U˚ D U
1
F C a.t/U
1
3 Pd U˚ DW F1
(3.92)
or, with ˚.0/ D Z.0/, i@ t ˚ C H0 ˚ D
U
1
.V
HPd C iıPd /U˚ C F1 :
(3.93)
127
3.4 Dispersive estimates for the perturbed linear evolution
Choose a smooth, exponentially decaying matrix potential V2 which is invertible and such that the operator V1 WD .V
1
HPd C iıPd /V2
is bounded from Lp ! Lq for any 1 p; q 1. In other words, V1 V2 D V
(3.94)
HPd C iıPd
with V1 ; V2 being bounded from Lp ! Lq for any 1 p; q 1. By Duhamel the solution to (3.93) is Z t i t H0 (3.95) ˚.t/ D e ˚.0/ i e i.t s/H0 Œ U 1 V1 V2 U˚ C F1 .s/ ds : 0
Applying U.t/ to both sides yields, since U commutes with the propagator of H0 , Z t i tH0 Z.t/ D U.t/e Z.0/ C i e i.t s/H0 U.t/U 1 .s/V1 V2 Z.s/ U.t/F1 .s/ ds: 0
(3.96)
We introduce the operators t
Z T0 F .t/ WD V2
e i.t
s/H0
V1 F .s/ ds;
e i.t
s/H0
U.t/U.s/
0
TQ0 F .t/ WD V2
t
Z
1
V1 F .s/ ds :
0
By the Strichartz estimates for the free equation, T0 ; TQ0 are bounded on L2t;x . By (3.96), Z t it H0 Q V2 Z D i T0 V2 Z C V2 U.t/e Z.0/ iV2 U.t/ e i.t s/H0 F1 .s/ ds : (3.97) 0
Suppose .Id
i TQ0 /
1
W L2t;x ! L2t;x
(3.98)
as a bounded operator. Then (3.97) implies via the endpoint Strichartz estimate, see (3.92), kV2 ZkL2 . kZ.0/k2 C kF1 kS . kZ.0/k2 C kF kS C c0 kV2 ZkL2 : t;x
t;x
128
3 Above the ground state energy I: Near Q
To pass to the final estimate we wrote Pd Z D Pd V2 1 V2 Z and used that Pd V2 1 is bounded by construction. Inserting the resulting bound on kV2 ZkL2 back into t;x (3.96) yields the desired estimate (3.91). It therefore remains to prove (3.98) which will follow from .Id
iT0 /
1
W L2t;x ! L2t;x
(3.99)
provided we can show that kT0 TQ0 k 1 in the operator norm on L2t;x . This, however, follows from the pointwise dispersive estimate on e itH0 which yields 1 5 4 ht si 4 kF .s/k2 : kV2 e i.t s/H0 U.t/U.s/ 1 1 V1 F .s/k2 . kak1 Thus, we have reduced ourselves to proving (3.99). We introduce Z t T1 F .t/ WD V2 e i.t s/HPc .t s/ıPd V1 F .s/ ds: 0
As for the meaning of T1 , first note that due to commutativity e itHPc
tıPd
D e it HPc e
tıPd
D e it H Pc C Pd e
tıPd
D e it H Pc C e
tıPd
Pd
satisfies Strichartz estimates as in (3.87), see [8] and [43]. Second, the solution to i@ t Z C HPc Z C iıPd Z D 0 can be written in two ways: Z.t/ D e itHPc
tıPd
Z.0/ ; Z t itH0 Z.t/ D e Z.0/ C i e i.t
s/H0
.V
HPd C iıPd /Z.s/ ds:
0
Thus, one further has e i t HPc
t ıPd
De
Z.0/
it H0
t
Z
e i.t
Z.0/ C i
s/H0
.V
HPd C iıPd /e isHPc
sıPd
Z.0/ ds :
0
Therefore, we conclude that Z tZ s T0 T1 F .t/ D V2 e i.t 0
s/H0
.V
HPd C iıPd /e i.s
s1 /HPc .s s1 /ıPd
0
V1 F .s1 / ds1 ds t
Z D
.e i.t
iV2 0
s1 /HPc .t s1 /ıPd
e i.t
s1 /H0
/V1 F .s1 / ds1
129
3.4 Dispersive estimates for the perturbed linear evolution
or T0 T1 C i.T1
T0 / D 0 which implies that iT0 /.Id C iT1 / D Id :
.Id On the other hand, s
Z tZ
e i.t
T1 T0 F .t/ D V2
s/HPc .t s/ıPd
.V
HPd C iıPd /e i.s
s1 /H0
0
0
V1 F .s1 / ds1 ds Z tZ
t
@s e i.t
D iV2 0
e i.s
s1 /H0
ds V1 F .s1 / ds1
s1 t
Z D
s/HPc .t s/ıPd
.e i.t
iV2
s1 /HPc .t s1 /ıPd
e i.t
s1 /H0
/V1 F .s1 / ds1
0
whence T1 T0 C i.T1
T0 / D 0 which implies that .Id C iT1 /.Id
iT0 / D Id :
These identities hold in the algebra of bounded operators on L2t;x , as justified by the endpoint Strichartz estimates. Thus (3.99) holds and (3.87) follows. For (3.88) one applies a gradient to (3.86). From (3.95), we obtain the scattering of ˚ in the following sense: ˚.t/ D e itH0 ˚1 C o.1/;
t !1
in H 1 for some ˚1 2 H 1 . Thus, Pc .t/ D Pc U.t/ e itH0 ˚1 C o.1/ D e iA.t/3 e it H0 ˚1 C o.1/;
t !1
in H 1 , as claimed. The proof with A ¤ 0 is left to the following exercise. Exercise 3.30. Adapt the previous proof to include the term iAr with A ¤ 0.
130
3 Above the ground state energy I: Near Q
3.5
The center-stable manifold for the radial cubic NLS in R3
In this section, we construct9 a center-stable manifold containing the ground state Q for the NLS equation (3.83). All function spaces will be radial. Moreover, Bı .Q/ denotes a ı-ball in the energy space centered at Q. We work with the matrix formalism as it appears in Section 3.4.4. In what follows, H.˛; / D
C ˛ 2 2Q2 .; ˛/ e 2i Q2 .; ˛/
e 2i Q2 .; ˛/ : ˛ 2 C 2Q2 .; ˛/
(3.100)
The following proposition constructs the center-stable manifold in a small neighborhood of Q. It should be compared to Definition 6.13; in fact, it provides much more detailed information than what is required by that definition. In view of the symmetries of (3.83) it is more natural to work with the “cone” of radial solitons S WD fe i Q.; ˛/ j 2 R; ˛ > 0g; rather than with a fixed soliton Q. Moreover, S˛ is the slice of S given by fixing ˛. In Remark 3.32 we therefore extend M so as to cover all of the radial soliton manifold S, and Corollary 3.33 characterizes the stable manifold, which lies in M. All of this is in analogy with the results which were obtained above for the NLKG equation. However, as we shall see, carrying out the underlying Lyapunov–Perron scheme is much more involved for the NLS equation. As we already mentioned several times before, the following proposition has its roots in the asymptotic stability analysis of solitons. For results on asymptotic stability analysis in the subcritical, and thus orbitally stable case, see Buslaev, Perelman [22], [23], Cuccagna [42], and Soffer, Weinstein [128]. See also Pillet, Wayne [117] for invariant manifolds in the context of the stability analysis of small solitons obtained by bifurcation off of linear eigenfunctions. 1 Theorem 3.31. There exists ı > 0 small and a smooth manifold M Hrad with the following properties: S \ Bı .Q/ M Bı .Q/, M divides Bı .Q/ into two connected components, and any initial data u0 2 M generates a solution of (3.83) for all t 0 of the form
u.x; t/ D e i.t/ Q x; ˛.t / C v.x; t/;
9
8t 0
Skip on first reading, this is not needed for the main theorems on the NLKG equation.
(3.101)
3.5 The center-stable manifold for the radial cubic NLS in R3
where .t/ D
Rt 0
131
˛ 2 .s/ ds C .t/, k P kL1 \L1 .0;1/ C k˛k P L1 \L1 .0;1/ . ı 2 ; supŒj˛.t/ t0
1j C j .t /j . ı :
(3.102)
The function v is small in the sense kvkL1 C kvkL2 ..0;1/IW 1;6 .R3 // . ı 1 3 t ..0;1/IH .R // t
(3.103)
and it scatters: v.t/ D e it v1 C oH 1 .1/ as t ! 1 for a unique v1 2 H 1 . M is unique in the following sense: there exists a constant C so that any u0 2 Bı .Q/ satisfies u0 2 M if and only if the solution u.t/ of (3.83) with data u0 has the property that dist.u.t/; S1 / C ı for all t 0. Proof. Inserting (3.101) into (3.83) yields ! ! v v e i@ t C H.t/ D .t/ P e .t/ v v where
e D H.t/
2Q2 .; ˛.t// e 2i.t/ Q2 .; ˛.t//
e .t; v; v/ i ˛.t/e P .t/ C N
(3.104)
e 2i.t/ Q2 .; ˛.t// C 2Q2 .; ˛.t//
(3.105)
as well as e .t/ D
! e i.t/ Q.; ˛.t// ; e i.t/ Q.; ˛.t//
! e i.t/ @˛ Q.; ˛.t// e .t/ D e i.t/ @˛ Q.; ˛.t//
(3.106)
and e .t; v; v/ D N
! 2e i.t/ Q.; ˛.t//jvj2 e i.t/ Q.; ˛.t//v 2 C jvj2 v : (3.107) 2e i.t/ Q.; ˛.t//jvj2 C e i.t/ Q.; ˛.t//v 2 jvj2 v
Next, set v.t/ D e
i0 .t/
t
Z w.t/;
˛ 2 .s/ ds:
0 .t/ D
(3.108)
0
Then (3.104) turns into i@ t W C H .t/W D .t/.t/ P
i ˛.t/.t/ P C N .t; W /
(3.109)
132
3 Above the ground state energy I: Near Q
where W D
w w
H .t/ D
, and with D .˛; /, C ˛ 2 .t/ 2Q2 .; ˛.t// e 2i .t/ Q2 .; ˛.t//
e 2i .t/ Q2 .; ˛.t// ˛ 2 .t/ C 2Q2 .; ˛.t//
(3.110)
as well as .t/ D
! e i .t/ Q.; ˛.t// ; e i .t/ Q.; ˛.t//
! e i .t/ @˛ Q.; ˛.t// .t/ D e i .t/ @˛ Q.; ˛.t//
(3.111)
and N .t; W / D
! 2e i .t/ Q.; ˛.t//jwj2 e i .t/ Q.; ˛.t//w 2 C jwj2 w : (3.112) 2e i .t/ Q.; ˛.t//jwj2 C e i .t/ Q.; ˛.t//w 2 jwj2 w
At this point we remark that all manipulations which we perform in this proof on f (3.104) and (3.109) preserve the “admissible” subspace f f j f W R3 ! Cg. This is necessary in order to return to the scalar formulation (3.101). In other words, the second row of these systems can be viewed as redundant, as it is always the complex conjugate of the first. 1 0 Let 3 D be the third Pauli matrix, and set .t/ D 3 .t/, .t/ D 0 1 3 .t/. Impose the orthogonality conditions10 ˝ ˛ ˝ ˛ W .t/; .t/ D 0; W .t /; .t/ D 0; 8t 0 : (3.113) Note that this imposes a condition on the data at t D 0. However, by the inverse function theorem there is a unique choice of ˛.0/ and .0/ in a ı-neighborhood of .1; 0/ so that (3.113) is satisfied; the needed nondegeneracy here is provided by hQj@˛ Qi ¤ 0. Since H .t/ .t/ D 0 and H .t/ .t/ D 2˛ .t/, as well as ˝ ˛ ˝ ˛ .t/; .t/ D .t/; .t/ D 0; ˝ ˛ ˝ ˛ ˝ ˛ .t/; .t/ D .t/; .t/ D 2 Qj@˛ Q ¤ 0; one obtains from (3.109) that ˝ ˛
.t/ P .t/; .t/ D ˝ ˛ i ˛.t/ P .t/; .t/ D 10
˝ ˛ i W .t/; P .t/ ˝ ˛ i W .t/; P .t/
˝ ˛ N .t; W /; .t/ ; ˝ ˛ N .t; W /; .t/ :
In this section, we write h; i for the standard inner product in L2 .R3 I C2 /.
(3.114)
3.5 The center-stable manifold for the radial cubic NLS in R3
133
The time derivatives P ; P .t/ on the right-hand side come with a small coefficient (due to the W ), and are therefore admissible in the contraction. The system (3.109), (3.113), (3.114) determines the evolution of v.t/; ˛.t/; .t/ in (3.101). It suffices for (3.113) to hold at one point, say t D 0, since it then holds for all t 0. We need to find a fixedpoint to this system consisting of a path .t/ D .˛.t/; .t// as well as a function vv , or equivalently, W satisfying the system as well as the bounds (3.102), (3.103). We begin with the stability part of the underlying contraction argument, i.e., we turn (3.102) and (3.103) into bootstrap assumptions and then recover them from this system. Thus, suppose 0 D .˛0 ; 0 / and W0 are given so that (3.102) and (3.103) hold and consider the following system of differential equations: i@ t W C H0 .t/W D .t/ P i ˛.t/ P 0 .t/ 0 .t/ C N0 .t; W0 /; ˝ ˛ ˝ ˛ ˝ ˛
.t/ P 0 .t/; 0 .t/ D i W .t/; P 0 .t/ N0 .t; W0 /; 0 .t/ ; ˝ ˛ ˝ ˛ ˝ ˛ i ˛.t/ P 0 .t/; 0 .t/ D i W .t/; P0 .t/ N0 .t; W0 /; 0 .t/ ; ˝ ˛ ˝ ˛ W .0/; 0 .0/ D 0; W .0/; 0 .0/ D 0;
(3.115)
where H0 , 0 ; 0 and N0 .t; W0 / are defined as above but relative to the given functions 0 ; W0 . The initial conditions are ˛.0/ D ˛0 .0/, .0/ D 0 .0/; in addition to the final equation in (3.115), W .0/ needs to satisfy a further codimension-1 condition which will be specified below. We begin with the ˛; P P part of (3.115). The W appearing on the right-hand side will be seen later to satisfy (3.103); for the moment, we will simply assume this bound. To be more specific, rewrite (3.102) and (3.103) in the form k k P L1 \L1 C k˛k P L1 \L1 C0 ı 2 kvkL1 1 3 C kvkL2 W 1;6 .R3 / C1 ı t H .R /
(3.116)
t
and assume that C0 C12 . Inserting these bounds in the right-hand side of (3.114) yields k k P L1 \L1 C k˛k P L1 \L1 . C0 C1 ı 3 C C12 ı 2 C0 ı 2 provided ı is small. One can thus recover (3.116). The bound on v (or W ) is more delicate. Since we are in the unstable regime, (3.109) is exponentially unstable. More precisely, write H0 .t/ D H0 C a0 .t/3 C D0 .t/
134
3 Above the ground state energy I: Near Q
with the constant coefficient operator H0 D H.˛0 .0/; 0 .0//, see (3.100), and a0 .t/ D ˛02 .t/ ˛02 .0/, as well as D0 .t/ equaling e 2i 0 .t/ Q2 ; ˛0 .t / C e 2i 0 .0/ Q2 ; ˛0 .0/ Q2 ; ˛0 .0/ 2 Q2 ; ˛0 .t / : 2 2 2i 0 .0/ 2 2i 0 .t/ 2 e
Q ; ˛0 .t /
2 Q ; ˛0 .t /
Q ; ˛0 .0/
e
2
N
Q ; ˛0 .0/
2
Note that ka0 ./k1 . ı and khxi D0 ./k1 . ı for any N provided the condition (3.102) holds. Proposition 3.28 in Section 3.4.4 details the spectral properties of H0 provided ˛.0/ D 1 and .0/ D 0. The more general case here follows by means of the rescaling f 7! ˛f .˛x/, as well as a modulation by a constant unitary matrix. Following the notation of Proposition 3.28 one writes W .t/ D C .t/GC C .t/G C W1 .t/
(3.117)
˝ ˛ where W1 .t/; 3 G˙ D 0 for all t 0. One needs to apply the aforementioned rescaling and modulationto G˙ and with the fixed parameters ˛0 .0/; 0 .0/, which means that D ˛0 .0/ , G˙ D G˙ ˛0 .0/; 0 .0/ . We remark that ˙ as defined in (3.117) are real-valued. Indeed, since kG˙ k2 D 1 and GC D ggC D G , the C Riesz projections associated with the eigenvalues ˙i can be seen to be P˙ D
h; 3 G i G˙ ; hG˙ ; 3 G i
(3.118)
where hG˙ ; 3 G i 2 i R n f0g. Therefore, C D
hW; 3 G i 2ihwjig i D 2R hG˙ ; 3 G i hG˙ ; 3 G i
and similarly for . We now rewrite the W -equation in (3.115) in the form i P C .t/GC C i P .t/G
iC .t/GC C i .t/G
C i@ t W1 .t/ C .H0 C a0 .t/3 /W1 D
D0 .t/W C .t/ P 0 .t/
a0 .t/3 GC
(3.119)
a0 .t/3 G
i ˛.t/ P 0 .t / C N0 .t; W0 / DW F .t/ :
Denote by P˙ , P0 the Riesz projections onto G˙ , and the zero root space, respectively. Note that these operators are given by integration against exponentially decaying tensor functions. Moreover, we write Pc D 1
PC
P
P0 D Preal
P0
(3.120)
3.5 The center-stable manifold for the radial cubic NLS in R3
135
for the projection onto the continuous spectrum. Applying the projections P˙ to (3.119) yields the system of ODEs P C .t/ C .t/ GC D ia0 .t/PC .3 W1 / iPC F .t/ ; (3.121) P .t/ C .t/ G D ia0 .t/P .3 W1 / iP F .t/ : For “generic” initial data C .0/ the solution C .t/ grows exponentially. However, there is a unique choice of initial condition that stabilizes C (i.e., ensures that it remains bounded) leading to the determination of the co-dimension one manifold. It is given by means of the suppose x.t/ P x.t/ D f .t/ following simple principle: with f 2 L1 .0; 1/ . Then x 2 L1 .0; 1/ iff Z 1 (3.122) 0 D x.0/ C e t f .t/ dt: 0
Thus, 1
Z 0 D C .0/GC C i
e
t
0
a0 .t/PC .3 W1 /.t/
PC F .t/ dt
is that unique choice. (3.123) has the following equivalent formulation Z 1 C .t/GC D i e .s t/ a0 .t/PC .3 W1 /.t/ PC F .t/ ds :
(3.123)
(3.124)
t
For .t/ we have the expression .t/G D e
t
t
Z .0/G C i
e
.t s/
a0 .t/P .3 W1 /.t/
0
P F .t/ ds : (3.125)
Via (3.118) one checks that ˙ as defined by these equations are real-valued. To determine the PDE for W1 D Preal W1 D Preal W , we write W1 D Pc W1 C P0 W1 D Wdisp C Wroot . Then i @ t Wdisp .t/C H0 Ca0 .t/3 Wdisp D Pc F .t/ a.t/Œ3 ; PC CP CP0 W1 : (3.126) The sought after solution ˛.t/; .t/; C .t/; .t/; Wroot .t/; Wdisp .t/
(3.127)
is now determined from the second and third equations of (3.115), from (3.124), (3.125), and (3.126). The root part is controlled by the orthogonality conditions hW; 0 i.t/ D hW; 0 i.t/ D 0;
8 t 0:
136
3 Above the ground state energy I: Near Q
The main technical ingredient for the dispersive control of (3.126) is the Strichartz estimate of Lemma 3.29. The existence of the solution (3.127) is not entirely trivial since the determining equations contain these functions linearly on the right-hand side. However, they occur with small coefficients which allows one to iterate or contract; we skip those details. The solution obeys the estimates (3.102), (3.103). While (3.102) has already been established in this fashion, (3.103) is obtained as follows. Assuming again (3.116), one concludes from (3.124) and (3.125) that k˙ kL1 \L2 . ı C .C0 C1 ı C C0 C C12 /ı 2 C1 ı provided ı is sufficiently small. Via Lemma 3.29 we conclude that kWdisp kS C1 ı where S is the Strichartz space in (3.103). Finally, now that the path ˛.t/; .t/ has been determined, as well as ˙ .t/, Wdisp .t/, the orthogonality conditions (3.113) determine Wroot which also satisfies kWroot kS C1 ı. From these estimates, we conclude (3.103) via bootstrap as claimed. The manifold is determined by (3.123) as a graph, once a fixed point .; W / D .0 ; W0 / is obtained. More precisely, for fixed ˛.0/; .0/ we prescribe initial conditions W .0/ 2 H 1 , kW .0/ kH 1 . ı for (3.115) such that P0 W .0/ D 0 as well as PC W .0/ D 0 where the projections are relative to H ˛.0/; .0/ . Such data are linearly stable. The condition (3.123) takes nonlinear corrections into account and modifies the data in the form W .0/ D W .0/ C h.0 ; W0 ; W .0/ /GC
(3.128)
ˇ ˇ where h.0 ; W0 ; W .0/ / D C .0/ is real-valued and satisfies ˇh.0 ; W0 ; W .0/ /ˇ . ı 2 . Since .0/ D 0 .0/ by construction, once we have found a fixed point, we can write h.0 ; W0 ; W .0/ / D h ˛0 .0/; 0 .0/; W .0/ where the latter is smooth in W .0/ in the sense of Fréchet derivatives. Moreover, the bound ˇ
ˇ ˇh ˛0 .0/; 0 .0/; W .0/ ˇ . W .0/ 2 1 H
(3.129)
will hold. This shows that (3.128) describes a codimension-3 manifold which is smoothly parametrized by W .0/ and tangent to the subspace of linear stability. To regain the two missing codimensions, we vary ˛0 .0/; 0 .0/ in a ı-neighborhood of .1; 0/. In other words, we let the dilation and modulation symmetries act on the codimension-3 manifold. Since these symmetries act transversely on the manifold
3.5 The center-stable manifold for the radial cubic NLS in R3
137
(for the same reason that allowed us to enforce (3.113) at t D 0 by modifying the data), we obtain a smooth codimension-1 manifold which will be parametrized by ˛0 .0/; 0 .0/; W .0/ 2 .1 ı; 1 C ı/ . ı; ı/ Bı where Bı is a ı-ball in H 1 . This is then the sought after M. Thus, one needs to find a fixed point for the system (3.115) via a contraction argument. The contraction argument is slightly delicate as it involves solving this system with two different but nearby given paths j0 , j D 0; 1 which therefore define different Hamiltonians via (3.110), and therefore also different orthogonality conditions (3.113). Note that phases of the form t˛ 2 and t ˛Q 2 diverge linearly if ˛ 2 ¤ ˛Q 2 . This make it necessary to employ a weaker norm than the one used in the previous stability argument, see (3.102), (3.103). In order to carry out the comparison between two solutions, we work on the level of (3.104) rather than with the aforementioned W -system. Thus, consider two paths j0 .t/ D ˛j0 .t/; j0 .t/ satisfying (3.102) and with 0.0/ .0/ D 1.0/ .0/, and the associated equations with Z D vv ej .t/Z D Pj .t/e i@ t Z C H j .t/
ej .t; vj0 ; vj0 / i ˛Pj .t/e j .t/ C N
(3.130)
ej , e ej .t; v; v/ are defined as in (3.105), for j D 0; 1, see (3.104). Here H j , e j and N 0 (3.106), (3.107) but relative to the paths j .t/. Moreover, the function vj0 are given and satisfy (3.103), and we impose the orthogonality conditions, see (3.113), ˝ ˛ ˝ ˛ Z.t/; 3e j .t/ D Z.t/; 3e j .t/ D 0;
8 t 0:
(3.131)
The initial conditions for the paths are 0 .0/ D 1 .0/ D 0.0/ .0/, whereas for Z0 ; Z1 one invokes (3.128) as follows: fix Z0.0/ 2 Bı .0/ so that P0 Z0.0/ D PC Z0.0/ D 0 and set Z0 .0/ D Z0.0/ C h 00 ; W00 ; W0.0/ GC ; (3.132) Z1 .0/ D Z0.0/ C h 10 ; W10 ; W0.0/ GC : This choice guarantees that (3.131) holds at t D 0. By the preceding stability analysis, (3.130) and (3.131) then define unique solutions .j ; Zj / satisfying (3.102) and (3.103). Differentiating (3.131) in combination with (3.130) yields the modulation equations stated in (3.114). Thus, we rewrite (3.130) in the form ej .t/Zj D i @ t Zj C H
iLj .t/Zj C Nj .t; vj0 ; vj0 /
(3.133)
138
3 Above the ground state energy I: Near Q
ej and the nonlinear term in (3.114). The linear term where Nj incorporates both N Lj .t/Zj is of finite rank and corank, and satisfies the estimates kLj .t/Zj kW k;p . kZj kH 1 jPj0 .t/j for any 1 p 1 and k 0. Combining this pointwise in time bound with (3.102) yields the full estimates on Lj .t/Zj . By construction, any solution of (3.133) which satisfies (3.131) at one point, say t D 0, satisfies (3.113) for all t 0. The difference R WD Z1 Z0 satisfies e 0 .t/R D i@ t R C H
iL1 .t/R N0 .t; v00 ; v 00 / C N1 .t; v10 ; v 01 / e 0 .t/ H e 1 .t//Z1 i.L1 .t/ L0 .t//Z0 DW F e C .H
(3.134)
whereas the difference of the paths D 1 0 is governed by taking differences of the third and fourth equations, respectively, in (3.115) for j D 1; 0. We estimate .R; / in the norm, with > 0 small, fixed, and to be determined, k.R; /kY WD ke
t
RkL1 2 C ke t ..0;1/IL /
t
k P L1 ..0;1// :
(3.135)
To render this a norm, one fixes .0/ D 0, say. Note that some measure of growth e 0 .t/ H e 1 .t/ and L1 .t/ L0 .t/ grow linearly in t. has to be built into k kY , since H Next, we perform the same modulation as above, i.e., # " Z t 0 e i0 .t/ 0 0 R; .t/ D .˛00 .s//2 ds : W .t/ WD 0 i00 .t/ 0 e 0 Denoting the matrix here by M0 .t/, W satisfies the equation e 0 .t/ C ˛ 0 .0/ 2 3 W D M0 F e: i@ t W C H 0
(3.136)
To obtain estimates on (3.136), we write e 0 .t/ C ˛ 0 .0/ 2 3 D H0 C a.t/3 C D.t/ H 0 0 0 with the constant coefficient operator H0 D H ˛00 .0/; 00 .0/ , see (3.100), and 2 2 ˛00 .0/ , as well as D.t/ equaling a.t/ D ˛00 .t/ " # 0 0 e
2 Q2 ; ˛00 .t / Q2 ; ˛00 .0/ e 2i 0 .t/ Q2 ; ˛00 .t / C e 2i 0 .0/ Q2 ; ˛00 .0/ 0 Q2 ; ˛00 .t / e 2i 0 .0/ Q2 ; ˛00 .0/ 2 Q2 ; ˛00 .t / Q2 ; ˛00 .0/
2i 00 .t/
:
One has ka./k1 . ı 2 and khxiN D./k1 . ı 2 for any N as before. At this point the analysis is similar to the one starting with (3.117). Indeed, writing once again W D C GC C G C Wroot C Wdisp
3.5 The center-stable manifold for the radial cubic NLS in R3
139
where the decomposition is carried out relative to H00 , one inserts this into (3.136) and proceeds as before. The two main differences from the previous stability analysis are as follows: (i) the stability condition (3.123) holds automatically here, since we know a priori that C remains bounded; indeed, we chose Z1 ; Z0 to each satisfy (3.123) whence (3.103) holds for each of these functions. (ii) the orthogonality condition (3.113) does not hold exactly in this form, since it is obtained by taking the difference of the orthogonality conditions satisfied by Z1 and Z0 . But this is minor, since the error one generates in this fashion is contractive. Applying the dispersive bound of Lemma 3.29 (here we need only the L2x part) to Wdisp yields via a term-wise estimation of the right-hand side of (3.134), Z t
R.t/ . ıe t .R0 R0 ; 0 0 / C ı
R.s/ ds 0 1 0 1 2 Y 2 0 (3.137) Z 1
.s t/
R.s/ 2 ds Cı e t
where Rj0 D
vj0 vj0
for j D 0; 1. Recall that the initial conditions for R are deter-
mined by (3.132). The final integral in (3.137) is a result of the C equation (3.124). Assuming .˛00 .0// > ı, Gronwall’s inequality implies
sup e t R.t/ . ı 1 .v 0 v 0 ; 0 0 / 2
t0
0
1
0
1
Y
as desired. Next, one estimates the equation with initial condition .0/ D 0. The conclusion is a bound of the form
ˇ ˇ
.R; / C ˇh. 0 ; W 0 ; W .0/ / h. 0 ; W 0 ; W .0/ /ˇ .v 0 v 0 ; 0 0 / 0 0 1 1 0 1 0 1 Y 0 0 Y which proves the desired contractivity. See [9] for more details on these estimates. Hence, one has a fixed point of (3.115) as well as a well-defined function h.00 .0/; W0.0/ /. This concludes the proof of the existence part. Next, we turn to scattering. In contrast to the previous analysis, we do not lin earize around H ˛.0/; .0/ , but rather around H ˛.1/; .1/ . Thus, consider the system (3.114), (3.113), (3.124), (3.125), (3.126) with a.t/ D ˛ 2 .t/ ˛ 2 .1/, F defined by (3.119), and D.t/ equaling 2 Q2 ; ˛.t / Q2 ; ˛.1/ e 2i .t/ Q2 ; ˛.t / C e 2i .1/ Q2 ; ˛.1/ : 2i .t/ 2 2i .1/ 2 2 2 e
Q ; ˛.t /
e
Q ; ˛.1/
2 Q ; ˛.t /
Q ; ˛.1/
(3.138)
Thus, a.t/; D.t/ ! 0 as t ! 1. This ensures the vanishing at t D 1 of the first three terms of F .t/ in (3.119). The fourth and fifth terms of F vanish in the
140
3 Above the ground state energy I: Near Q
L1 .T; 1/-sense as T ! 1 by (3.102), whereas the nonlinear term N.t; W / vanishes in the sense of Strichartz estimates. Therefore, (3.124), (3.125) imply that ˙ .t/ ! 0 as t ! 1. Hence, in view of the scattering statement in Lemma 3.29 one has the representation in H 1 Wdisp .t/ D e i3
Rt
C˛ 2 .1/Ca.s// ds
D e i3
Rt
C˛ 2 .s// ds
0. 0.
W1 C o.1/;
(3.139)
W1 C o.1/
as t ! 1. The modulation in (3.108) removes the ˛ 2 .s/ in the exponent once we return to the v representation. Finally, by the orthogonality conditions Wroot .t/ ! 0. In summary, we have obtained the desired scattering statement for v in (3.101). Finally, to obtain the uniqueness statement let u.t/ be a solution with u.0/ 2 Bı .Q/ and with the property that dist.u.t/; S1 / . ı for all t 0. We claim that there exists a C 1 -curve ˛.t/; .t/ 2 1 O.ı/; 1 C O.ı/ R which achieves ˝ ˛ u.t/ e i.t/ Q ; ˛.t/ je i.t/ Q ; ˛.t/ D 0; (3.140) ˝ ˛ u.t/ e i.t/ Q ; ˛.t/ jie i.t/ @˛ Q ; ˛.t/ D 0 for all t 0, as well as
sup u.t/ t0
e i.t/ Q ; ˛.t/ H 1 . ı :
(3.141)
Q 2 R so that In fact, by definition there is a C 1 –path .t/
Q sup u.t/ e i .t/ Q.; 1/ H 1 . ı : t0
This shows that one can fulfill (3.140) up to O.ı/. Next, one uses that jhQj@˛ Qij ' Q can be mod1 for all ˛ ' 1 and the inverse function theorem to show that .˛Q 1; / ified by an amount O.ı/ so as to exactly satisfy (3.140) without violating (3.141). Furthermore, by chaining one concludes that this procedure yields a well-defined path .˛; / which is C 1 , as claimed. Next, define Z t 0 .t/ D ˛ 2 .s/ ds C .0/ 0
and set D
0 . Now write u.t/ D e i.t/ Q ; ˛.t/ C v.t/ D e i.t/ Q ; ˛.t/ C e i0 .t/ w.t/ :
(3.142)
This then allows one to rewrite (3.140) in the form ˇ ˇ he i .t / Q ; ˛.t/ ˇw.t/i D 0; hie i .t/ @˛ Q ; ˛.t/ ˇw.t/i D 0 :
(3.143)
3.5 The center-stable manifold for the radial cubic NLS in R3
141
w As before, consider W D w , and perform the decomposition (3.117). Inserting (3.142) into (3.83) yields, cf. (3.104), ! ! v v P C ˛ 2 .t/ e e e .t; v; v/ (3.144) .t/ i ˛.t/e P .t/ C N C H.t/ D .t/ i@ t v v e e , and N e are as in (3.105), (3.106), (3.107). Furthermore, with W D where H; ;e w , w i@ t W C H.t/W D .t/.t/ P i ˛.t/.t/ P C N.t; W / (3.145) see (3.109), (3.111), (3.112). The orthogonality conditions (3.143) are of the form ˝ ˛ ˝ ˛ W .t/; .t/ D 0; W .t /; .t/ D 0 (3.146) which is identical with (3.113). This places us in the exact same position that we started from in the existence proof. Thus, the decomposition (3.142) is such that (3.102) and (3.103) hold. The only difference here is that we know a priori that C .t/ is bounded. However, (3.122) guarantees that therefore (3.123) holds which forces the solution to lie on M as desired. Remark 3.32. Denote the manifold constructed in Theorem 3.31 by M1;0 . The same construction can be applied to e i Q.x; ˛/ instead of Q for any 2 T D R=2 Z and ˛ > 0, yielding a codimension 1 manifold in the phase space H which we denote by M˛; . By the uniqueness part of Theorem 3.31 one concludes that [ MS WD M˛; (3.147) ˛>0; 2T
is again a smooth manifold, which contains all of S. 3.31 By the proof of Theorem it is smoothly parametrized by ˛.0/;
.0/; W .0/ where P ˛.0/;
.0/ W .0/ D 0 0 and PC ˛.0/; .0/ W .0/ D 0 and W .0/ needs to be small enough. Looking ahead to the full dynamical description of (3.83) given in the final chapter, we can make the following remarks: MS has the property that any u0 2 MS leads to a solution of (3.83) defined on t 0 which scatters to S as t ! 1 in the sense of Definition 6.13. We emphasize that this is not the manifold .5/ [ .7/ [ .9/ appearing in Theorem 6.14. Rather, that manifold is the maximal backward evolution of MS under the NLS flow. Note that MS , thus extended by the nonlinear flow, is again a manifold. The following characterization of the stable manifolds will be needed in the proof of Theorem 6.15. It precisely captures the situation where the radiation part
142
3 Above the ground state energy I: Near Q
(i.e., the difference between u.t/ and the soliton in (6.70)) has vanishing scattering data and is therefore uniquely captured by .0/. Corollary 3.33. Let MS be as in (3.147). Suppose u0 2 MS with M.u0 / D M.Q/ forward scatters to S in the sense of Definition 6.13 so that (6.70) holds with u1 D 0. Then the solution u.t/ of (3.83) with data u0 approaches a soliton trajectory in S1 exponentially fast. Moreover, the solution is uniquely characterized by 1 2 S 1 and a real number 0 with j0 j . ı. The case where u is an exact soliton is characterized by 0 D 0. Proof. This follows from the construction carried out in the proof of Proposition 3.31, but with H0 D H.˛.1/; .1// as the driving linear operator; see that part of the proof dealing with scattering. By (6.71), ˛.1/ D 1. In fact, consider the representation W D C GC C G C Wroot C Wdisp relative to this choice of H0 , and solve the system (3.114), (3.113), (3.124), (3.125), (3.126) with a.t/ D ˛ 2 .t/ ˛ 2 .1/, F defined by (3.119), and D.t/ given by (3.138). For (3.114) one assigns the terminal conditions ˛.1/ D ˛1 D 1,
.1/ D 1 , for (3.125) we impose the initial conditions .0/ D 0 , and (3.126) is solved with scattering data W1 D 0, cf. (3.139). Note that C does not require any further data, see (3.124). Similarly, Wroot is determined by (3.113). The point is that we can solve the aforementioned system for ˛.t/; .t/ , and W .t/ satisfying (3.102), (3.103) by contracting in the strong norm
.W; / WD e t W 1 C e t P L1 ..0;1// Y L ..0;1/IL2 / t
(3.148)
for suitably chosen and small > 0. Note the contrast to (3.135). In the setting of (3.135) the exponentially decaying weights forced us to start from t D 0 when carrying out the contraction argument. In the case of (3.148), however, we can solve for Wdisp from t D 1 due to the exponentially growing weights. It is essential, though, that for we can still start at t D 0; this is due to the fact that equation (3.125) contains exponentially decreasing functions (one therefore needs < but nothing else). In summary, .t/ .1/ decreases exponentially, as do a.t/; D.t/, ˙ , Wdisp , Wroot . This proves the exponential approach to S1 . Since ˛ 2 .t/ ˛ 2 .1/ ! 0 and .t/ .1/ ! 0 at an exponential rate, u.t/ in fact converges to a soliton trajectory in S1 exponentially fast. The case of an exact soliton is given by W D 0, which the contraction argument characterizes as .0/ D 0 D 0.
3.6 Summary and conclusion
3.6
143
Summary and conclusion
In this chapter we introduced the two commonly used methods in the construction of invariant manifolds, namely the Hadamard (graph transform) on the one hand, and the Lyapunov–Perron method on the other hand. Both approaches apply to infinite dimensions, and in either case the analysis is perturbative around the equilibrium state. The former has the advantage of requiring less spectral information on the linearized operator, whereas the latter allows for an asymptotic description of solutions starting on the center-stable manifold in forward time (they scatter to the equilibrium in a suitable sense). In order to obtain this asymptotic description, strong dispersive control is required from the linearized flow which can be difficult to establish. Both the Hadamard and the Lyapunov–Perron methods apply to the case where the equilibrium is subject to modulation as a result of symmetries. For the Lyapunov–Perron method we demonstrated this here by means of the radial NLS equation where one encounters both the modulation and dilation symmetries. For the graph transform, we did not include the modulation theory in this chapter since it requires ideas that go beyond the original Bates and Jones proof [6]. However, details can be found in [112] which extends the treatment of the NLKG equation by the Hadamard method to all energy subcritical powers of the nonlinearity as well as general nonradial energy data.
4
Above the ground state energy II: Moving away from Q
In contrast to the previous chapter which focused on the stable behavior of solutions near the ground states .˙Q; 0/, this chapter describes those solutions that enter, but then again leave, a neighborhood of the ground states. These are the non-trapped trajectories. Of particular importance here will be the fact that E.E u/ < J.Q/ C "2 where " > 0 is much smaller than the size of the neighborhood in which Theorem 3.22 guarantees the existence of the center-stable manifolds. Moreover, we will only describe the dynamics of non-trapped trajectories after they exit a ball of size C " where C is sufficiently large (in fact, relative to a suitable metric, called the nonlinear distance function below, we shall be able to take C D 2). As it turns out, after the exit from the C "-ball the dynamics is dominated by the unstable manifold which means that the coordinate in that direction grows exponentially. Inside the C "-ball around .˙Q; 0/ the dynamics can be very complicated relying on a delicate interplay between the different modes (i.e., between the exponential behavior and the dispersive part of the equation). We treat this region as a “black box” which we do not analyze at all. In fact, this is not needed as we are either trapped or being ejected out of that ball, and then carried to much larger (albeit still small compared to 1) distances from the equilibria. It is worth noting that Strichartz estimates do not enter into the analysis of the ejection mechanism itself. Rather, we will control the dispersive part via a suitable energy estimate. Furthermore, throughout this text, Strichartz estimates for the linearized operator LC , as they appeared in the previous chapter, are needed exclusively for the scattering property on the center-stable manifolds as constructed by the Lyapunov–Perron method.
4.1
Nonlinear distance function, eigenmode dominance, ejection
This section presents the process by which solutions that do not remain close to the ground state for all positive times are ejected from any small neighborhood of it after some positive time. Of particular importance for the description of the global dynamics are the signs of K0 and K2 as the solution exits a fixed small neighborhood of ˙.Q; 0/, as well as the following two facts:
146
4 Above the ground state energy II: Moving away from Q
(i) (ii)
the solution cannot return to that neighborhood the sign of K0 , K2 can only change if the solution reenters that neighborhood. Fact (i) constitutes the one-pass (or no-return) theorem which we prove in Section 4.3, whereas (ii) is a variational property which we prove in this section. The proofs of both properties require that the energy exceeds that of Q only by a slight amount. Once we have established (i) and (ii), we can deduce the following properties about the trajectories of NLKG (in forward time, say): if uE enters a small ball of fixed size ı0 (much larger than " and determined such that the signs of K0 ; K2 are constant outside that ı0 -ball) then either the solution is trapped in that ball or not. The former case occurs if and only if the solution falls on the manifold M of Theorem 3.22, whereas in the latter case it is ejected from the ball in such a way that the signs of K0 ; K2 are determined once and for all at exit time; in fact, M divides the ball into two halves and each half corresponds to a fixed sign. By the one-pass theorem, we then conclude that the sign of K0 ; K2 does not change ever again after the exit time. But this allows us to revert to the Payne, Sattinger arguments to conclude either global existence or finite time blowup (with the scattering to zero being again somewhat more difficult). We begin the technical part of this section with a preliminary analysis of the unstable dynamics near .˙Q; 0/. As in the previous chapter, all functions are radial and real-valued.
4.1.1 A first look at the ejection process Writing u D Q C v, we begin by recalling the expansions (3.2)–(3.4), i.e., vR C LC v D 3Qv 2 C v 3 D N.v/; ˛ 1 1˝ 3 P 22 C O kvkH E.Q C v; v/ P D J.Q/ C LC vjv C kvk 1 ; 2˛ 2 ˝ 2 K0 .Q C v/ D 2 Q3 jv C O kvkH 1 : With > 0 the L2 -normalized ground state of LC , it is now natural to write v.t; x/ D .t/.x/ C .t; x/;
? :
(4.1)
4.1 Nonlinear distance function, eigenmode dominance, ejection
147
Indeed, (4.1) becomes (with P N.v/ D N .v/) R
k 2 D N .v/;
R C LC D P? N.v/; ˛ 1 1˝ 1 3 P 22 C O kvkH E.Q C v; v/ P D J.Q/ C .P 2 k 2 2 / C LC j C k k 1 ; 2 2 2 ˝ ˛ 2 K0 .Q C v/ D 2 Q3 j C C O kvkH 1 : (4.2) From Chapter 2 we know that the sign of K0 (and K2 ) determines the fate of the solution, albeit only in the regime E.u; u/ P < J.Q/ D E.Q; 0/. As one would expect, this property remains true also if E.u; u/ P < J.Q/ C "2 D E.Q; 0/ C "2
(4.3)
dist .u; u/; P .Q; 0/ "
(4.4)
provided
where the distance is in H. We prove this in Lemma 4.8 below. In order to exploit this fact, we shall need to make sure that at exit time from a ı-ball Bı D Bı .Q; 0/ one has jj k kH 1 . Indeed, if this is not the case then the sign of K0 .Q C v/ cannot be controlled, see (4.2). On the other hand, if we can insure this property then it would agree with our expectation that the center-stable manifold of the previous chapter divides Bı into two halves: the one where the solution exits with > 0 and the other where it does so with < 0. Provided the one-pass property (i) from above holds, then we can indeed prove that the two halves correspond to global existence and blow-up, respectively (in forward time). The energy constraint (4.3) gives some indication of the -dominance. Indeed, in connection with (4.2), (4.3) implies that 2 P 2 C k kH P k22 . 2 C "2 : 1 C k
(4.5)
Thus, if (4.4) holds, then dist .u; u/; P .Q; 0/ ' jj " which, however, is insufficient in order to control K0 . Nevertheless, the desired conclusion can be reached by means of finer estimates on (4.2). First, set ! WD 1 .P? LC / 2 . By Exercise 3.3, k!f k2 ' kf kH 1 for any f ? . Then 1 E.Q C v; v/ P D J.Q/ C .P 2 2
k 2 2 / C
1 3 (4.6) k k P 22 C k! k22 C O kvkH 1 2
148
4 Above the ground state energy II: Moving away from Q
as well as K0 .u/ D h u C u
u3 jui
v 3 jQ C vi 2 D hLC vjQi C O kvkH 1 D hLC v D
3Qv 2
(4.7)
2 k 2 hjQi C h! j!Qi C O kvkH 1 :
From (4.6), k! k22 k 2 2 C C "2 . Thus, the second term in (4.7) can be estimated as follows: ˇ ˇ ˇh! j!Qiˇ2 k! k2 k!Qk2 hP ? LC QjQi.k 2 2 C C "2 / 2 2 hLC QjQi hjQihjLC Qi .k 2 2 C C "2 / hjQi2 k 2 2kQk44 .k 2 2 C C "2 / which implies that for some absolute constant c > 0, jh! j!Qij k 2 jj hjQi
c C O."2 =/
provided " jj 1. Inserting this bound into (4.7) leads to the desired conclusion, viz. jK0 .Q C v/j ' jj; sign K0 .Q C v/ D sign ./ provided " jj 1 : (4.8) The derivation of (4.8) is arguably somewhat delicate and non-robust; in fact, we remark that it is not clear how to repeat this argument in the context of the NLS equation. Therefore, it is perhaps more natural to seek a dynamical argument based on the robust idea that a solution starting out in a small ball around the equilibrium and which is ejected out of a much larger ball around that point should do so by means of the exponentially expanding (i.e., unstable) dynamics. In passing, we remark that an “onion” structure around .Q; 0/ represented by balls of different radii will be crucial for our argument. Note, however, that dynamically speaking is not meaningful by itself; since the P i.e., the phase space -equation in (4.2) is second order one has to consider .; /, variables. In order to distinguish between the expanding and contracting modes, we will now change coordinates in this two-dimensional space as follows: the homogeneous equation R k 2 D 0
4.1 Nonlinear distance function, eigenmode dominance, ejection
149
KD0 Wu
K>0 K 0). This suggests setting 1P 1 . C / 2 k 1 1P WD . / 2 k
C WD
(4.9)
since 0 is the same as .t/ D 0 e ˙kt . Moreover, D C C ;
P D k.C
/
(4.10)
and the ODE for in (4.2) becomes the equivalent system 1 N .v/ ; 2k 1 P C D kC C N .v/ : 2k Moreover, the energy expansion in (4.2) reads P D
k
(4.11)
1 1 3 P 22 C O kvkH 2k 2 C C hLC j i C k k 1 : (4.12) 2 2 Equations (4.11), (4.12) and the equation E.Q C v; v/ P D J.Q/
R C LC D P? N.v/
(4.13)
150
4 Above the ground state energy II: Moving away from Q
now allow for the following conclusions. First, we seek a criterion on the data by which the corresponding solution is guaranteed to be ejected from a neighborhood of .Q; 0/. One natural possibility is to require that at time t D 0 one has ˇ ˇ ˇ ˇ ˇC .0/ˇ & ˇ .0/ˇ : (4.14) If the ejection takes place along the unstable manifold, then (4.14) will hold eventually. Moreover, note that (4.14) and (4.12) imply that
2 ˇˇ ˇ
k 1 C
P 2 . jC .0/ˇˇ .0/ˇ C "2 C ı 3 (4.15) 0 H 2
ˇ ˇ
' ˇC .0/ˇ ' ı0 ". This shows that, in particular, where 1 .v; v/.0/ P H (4.14) excludes that the data belong to the center-stable manifold, see Theorem 3.22. If we now ignore the nonlinear terms in (4.11) and (4.13), then it follows that C .t/ D C .0/e kt ;
.t/ D .0/e sin.!t/
.0/ P :
.t/ D cos.!t/ .0/ C !
kt
;
This implies that C .t/ dominates for all t 0 in the sense that
ˇ ˇ
.v; v/.t/
' ˇC .t/ˇ 8 t 0 : P H
(4.16)
(4.17)
In the following exercise, the reader is asked to verify that the linear dynamics gives the correct leading order up until the time where the solution is no longer 1. Exercise 4.1. (a) Under the condition (4.14), prove that for given " ı0 ı1 1 the so is strictly increasing lution v.t/; v.t/ P has the property that v.t/; v.t/ P H for 0 < t0 < t < t1 where t0 ; t1 are constants (t0 depends on the implicit constant in (4.14)). In particular, there exists a unique time t 2 .t0 ; t1 / with
v.t /; v.t P / H D ı1 . Moreover, prove that jC .t /j ' ı1 ; k .t /kH 1
j .t /j ' ı0 C O.ı12 / ;
C
.t P / 2 ' ı0 C O.ı12 / ;
(4.18)
and conclude that K0 .u.t // ' C .t / ' .t /, as desired. Hint: for the
-part, use energy estimates. Note that this only works if the power of the nonlinearity does not exceed 3.
4.1 Nonlinear distance function, eigenmode dominance, ejection
151
ˇ ˇ ˇ ˇ (b) Now assume that ˇC .0/ˇ ˇ .0/ˇ. Prove that on some time interval 0 t t2 where t2 is a constant that depends on the implicit constant in the condition
is strictly decreasing. on the data, one has that v.t/; v.t/ P H (c) Repeat this exercise for nonlinearities jujp 1 u with powers 3 < p < 5. You need to assume the gap property (3.12) in that case. While the result of Exercise 4.1 is satisfactory in many ways, the method of proof is not robust in the following sense: while energy estimates were sufficient for p D 3, for powers 3 < p < 5 one needs to use Strichartz estimates for the KG-evolution with LC instead of C 1. As we emphasized before, this requires the gap property (3.12) for LC . Therefore, the reliance on Strichartz estimates for the -part is somewhat heavy-handed. Indeed, in this regime which is dominated by the hyperbolic dynamics (i.e., by the exponentially growing mode) one would not expect to encounter a subtle analysis of a subordinate component of the evolution such as . Following [109], we show in the following section how to avoid any dispersive analysis of by means of a combination of the energy and . The key quantity in this process is the nonlinear distance function to which we now turn.
4.1.2 Nonlinear distance function We begin by introducing the nonlinear distance function relative to the ground states ˙Q. Let u D ŒQ C v;
v D C ;
?
(4.19)
for D ˙, and where v is small. Define the linearized energy as 1 2 2 2 k hvji2 C k!P? vkL P L 2 C kvk 2 2 1 2 2 2 P 2 C k.! ; /k D k jj C jj P L 2 L2 2
kE v k2E WD
(4.20)
where ! is as above. By Lemma 3.2, 2 2 kE v k2E ' kE v k2H D kvkH P L 1 C kvk 2:
The relevance of k kE lies with the following decomposition, which follows from (4.6): E.E u/
J.Q/ C k 2 2 D kE v k2E
C.v/;
4 C.v/ WD hQjv 3 i C kvkL 4 =4:
(4.21)
152
4 Above the ground state energy II: Moving away from Q
There exists 0 < ıE 1 such that ˇ ˇ 2 kE v kE 4ıE H) ˇC.v/ˇ vE E =2:
(4.22)
Let be a smooth function on R such that .r/ D 1 for jrj 1 and .r/ D 0 for jrj 2. We define q v kE =.2ıE / C.v/: v k2E kE d .E u/ WD kE It has the following properties kE v kE =2 d .E u/ 2kE v kE ;
d .E u/ D kE v kE C O kE v k2E ;
d .E u/ ıE H) d2 .E u/ D E.E u/
J.Q/ C k 2 2 :
(4.23)
Note that the right-hand side of (4.23) is entirely controlled by since the other two terms are constant. To illustrate how one could hope to control the -component via , note the following heuristic argument. First, assume that the nonlinearity N.v; Q/ vanishes identically. Then E.E u/
J.Q/ D
1 2
k 2 2 C P 2 C k.! ; /k22 :
(4.24)
By (4.2) one has @ t . k 2 2 C P 2 / D 0 (assuming N D 0). Therefore, by (4.24), @ t k.! ; /k22 D 0 which is nothing but energy conservation for the -equation. This observation is robust enough to allow for a higher-order perturbation such as N.Q; v/, see the proof of Lemma 4.3. Henceforth, we shall always assume that uE is decomposed as in (4.19) such that dQ .E u/ WD inf d˙ .E u/ D d .E u/; ˙
where the choice of sign is unique as long as dQ .E u/ 2ıE . Recall from the previous section that ˙ .t/ WD
1 P .t/ ˙ .t/=k 2
are the unstable/stable modes for t ! 1 relative to the linearized hyperbolic evoluE but which are moving tion. First, we investigate the solutions which are close to ˙Q, away from these points.
4.1 Nonlinear distance function, eigenmode dominance, ejection
153
4.1.3 Eigenmode dominance By means of the nonlinear distance function we can now express estimate (4.5) very succinctly. Lemma 4.2. For any uE 2 H satisfying 2 E.E u/ < J.Q/ C dQ .E u/=2;
dQ .E u/ ıE ;
(4.25)
one has dQ .E u/ ' jj. In particular, has a fixed sign in each connected component of the region (4.25). Proof. (4.23) yields 2 dQ .E u/ D E.E u/
2 J.Q/ C k 2 2 < dQ .E u/=2 C k 2 2 :
2 and so, k 2 2 =16 kE v k2E =8 dQ .E u/=2 < k 2 2 . Inside of the set (4.25) one can never have D 0, since that would mean both dQ .E u/ D 0 and E.E u/ < J.Q/ which is impossible.
4.1.4 Ejection process The following ejection lemma is the key to extracting the hyperbolic nature from our PDE. It replaces the ejection process described in Exercise 4.1 by one which does not rely on any dispersive analysis of . The key condition (4.14) is replaced with (4.27). Note that it excludes the scenario described in Exercise 4.1, Part (b), and therefore implies (4.14). However, we reiterate that the following result does not depend on the gap property 3.12 and is a fairly robust statement; indeed, the NLS equation allows for an analogous one, see [110]. Lemma 4.3. There exists a constant 0 < ıX ıE with the following property. Let u.t/ be a local solution of (2.1) on an interval Œ0; T satisfying R WD dQ uE .0/ ıX ;
E.E u/ < J.Q/ C R2 =2
(4.26)
.0 < 8t < t0 / :
(4.27)
and for some t0 2 .0; T /, dQ uE .t/ R
154
4 Above the ground state energy II: Moving away from Q
ˇ d Alternatively, assume that dt dQ uE .t/ ˇ tD0 0. Then dQ uE .t/ increases monotonically until reaching ıX , and meanwhile, dQ uE .t/ ' s.t/ ' sC .t/ ' e kt R; 2 j .t/j C k .t/k E E .t/ ; E . R C dQ u (4.28) min sKs u.t/ & dQ uE .t/ C dQ uE .0/ ; sD0;2
with a fixed sign s D C1 or s D
1, where C 1 is an absolute constant.
Proof. Note that R > 0. Lemma 4.2 yields dQ .E u/ ' jj as long as R dQ .E u/ ıE , whereas the energy conservation of NLKG and the equation of give as long as dQ .E u/ ıE , see (4.20), 2 P .E u/ D 2k 2 ; @ t dQ
2 P 2 C 2k 4 jj2 C 2k 2 N .v/: .E u/ D 2k 2 jj @2t dQ
(4.29)
2 2 The exiting condition (4.27) implies @ t dQ .E u/j tD0 0. Since N .v/ . kvkH 1 , we 2 2 2 u/ as long as dQ .E u/ ' jj 1. u/ ' dQ .E have @ t dQ .E Hence, imposing ıX ıE and small enough, we deduce that dQ .E u/ R strictly increases until it reaches ıX ; meanwhile, dQ .E u/ ' s for s 2 f˙1g fixed. Since
2C
P 2 D =k 0;
we also infer that C ' . Next, integrating the equation (4.11) for C and using the C -dominance yields Z t Z t ˇ ˇ2 ˇ ˇ ˇ ˇ k.t s/ ˇ ˇC .t/ .0/ .t/ˇ . ˇ e k.t s/ ˇC .s/ˇ ds; N v.s/ ds . e C 0
0
kt where .0/ denotes the linearized solution. This implies both C .t/ D C .0/e
ˇ ˇ ˇ dQ uE .t/ ' ˇC .t/ˇ . Re kt and ˇC .t/
ˇ 2 2kt 2 ˇ .0/ ' dQ uE .t/ C .t/ . R e
whence also dQ .E u.t// ' jC .t/j ' R e kt for all t 0 up until the exit time from the ıX -ball. By the same argument, 2 j .t/j . R C dQ uE .t/ as claimed, which concludes the hyperbolic part. To bound the dispersive part , we do not rely on any dispersive estimates but rather on energy conservation. This becomes most transparent for the case of N D 0
4.1 Nonlinear distance function, eigenmode dominance, ejection
155
and C D 0, in other words when there are no higher order corrections. To be more specific, in absence of nonlinear corrections, energy conservation takes the form d 1 dt 2
k 2 2 .t/ C P 2 .t/ D 0;
2 d
.! ; / P L2 L2 D 0; dt
see (4.24). The desired estimate on will be a “nonlinearly perturbed" version of the latter vanishing. To be precise, from the equation, see (4.21), one has ˇ ˇ ˇ ˇ ˇ@ t Œ k 2 2 =2 C P 2 =2 C./ˇ D ˇ N .v/ N ./ P ˇ . k kH 1 jj2 : Subtracting it from the energy (4.21) yields ˇ ˇ ˇ@ t k k E 2E C.v/ C C./ ˇ . k kH 1 jj2 : Integrating this bound and using the bound on and .0/, E one obtains 2 2 E L1 R2 e 2kT ; k E kL 1 E.0;T / . R C k k t E.0;T / t
which implies the desired estimate for . Finally, recall from (3.4) that K0 .u/ D
k 2 hQji
2 h2Q3 j i C O kvkH 1 ;
and similarly we can expand K2 around Q: K2 .u/ D
.k 2 =2 C 2/hQji
2 h2Q C Q3 j i C O kvkH 1 :
(4.30)
Since hQji > 0 by their positivity, we obtain the desired bound on Ks . It is very simple to construct solutions which obey the ejection condition (4.27): simply choose initial conditions of the form .0/ D 0, .0/ E D 0, and C .0/ D R. Then the analysis of the .C ; ; / E system appearing in the previous proof shows that C .t/ grows exponentially and ensures that dQ .E u.t// grows initially, as well as up until the time that the scale ıX is attained. More generally, we can formulate the following corollary. ˇ ˇ ˇ ˇ Corollary 4.4. Suppose uE .0/ 2 H satisfies (4.26) as well as ˇC .0/ˇ ˇ .0/ˇ. Then (4.27) holds and thus the ejection lemma applies. Alternatively, assume that ˇ ˇ
ˇ ˇ ˇ .0/ˇ C
.0/ E H 1 L2 ˇC .0/ˇ ıX : Then both (4.26) and (4.27) hold, and thus the ejection lemma applies again.
156
4 Above the ground state energy II: Moving away from Q
ıx 2
Figure 4.2. A forbidden trajectory
Proof. The first assertion follows simply from ˇ ˇ 2 P @ t ˇ dQ .E u/ D 2k 2 .0/ D 2k 3 C .0/2 tD0
.0/2 0:
The second assertion follows from the first part, as well as the expansion of the energy to ensure (4.26). We leave the details to the reader. We remark that (4.27) is very natural from the point of view of the ensuing dynamical analysis. To be more specific, suppose a solution defined on some time interval I , and which satisfies (4.3), enters a ı-ball around .Q; 0/ where 0 < ı ıX . Further, assume that it is not trapped by the ıX -ball in forward time. Then there exists some time t 2 I at which " R D dQ uE .t / ıX . Taking t to be maximal with this property in I we conclude that (4.27) holds. Note that it is important here that " R. Indeed, the latter condition insures that is of smaller size than which is of course essential for the ejection process (the other extreme would be the center manifold). For later purposes we now exclude circulating trajectories, as in Figure 4.2. Lemma 4.5. There does not exist a solution to (2.1) with E.E u/ < J.Q/ C "2 and the following properties: u exists for all t 0 and 2" < dQ uE .t/ < ıX for all t 0.
4.1 Nonlinear distance function, eigenmode dominance, ejection
157
Proof. Let ı1 WD inf dQ uE .t/ : t0
We consider the following two cases: Case 1: uE attains ı1 . In fact, suppose that dQ uE .t0 / D ı1 for some t0 0. Then we can apply Lemma 4.3 starting at t0 to conclude that ıX is reached after some finite time contrary to our assumptions. Case 2: uE does not attain ı1 . Since dQ uE .t/ also cannot achieve any local minimum, again by Lemma 4.3, it must be monotonically decreasing. In other words, @ t dQ uE .t/ 0. Clearly, we need to also have @ t dQ uE .t/ ! 0 as t ! 1. On the other hand, it follows from (4.29) that @2t dQ uE .t/ & "2 for all t 0 since jj ' dQ .E u/ > 2". But these estimates are incompatible, and we are done. Another way of stating Lemma 4.5 is given in Corollary 4.7. It is in this form that we shall use it, and which is based on the following terminology. Definition 4.6. In what follows, we shall use the following terminology: ı a trajectory uE .t/ is trapped by an R-ball if it exists for all t 0 and if dQ uE .t/ R for all t T where T > 0 is some finite time. ı We also say that uE .t/, locally defined on some interval Œ0; T / is ejected from the ıX ball, if there exists a time interval Œt0 ; t1 Œ0; T / so that dQ uE .t/ is strictly increasing on Œt0 ; t1 and satisfies 1 2 uE .t0 / ; E.E u/ < J.Q/ C dQ 2 ıX dQ uE .t0 / D ; dQ .E u t1 / D ıX ; 10 dQ uE .t/ ' dQ uE .t0 / e k.t t0 / t0 < 8 t < t1 ; Ks u.t/ ' sign .t/ dQ uE .t/ t0 < 8 t < t1
(4.31)
for s D 0; 2. The implicit constants appearing in the following corollary are absolute. Corollary 4.7. Suppose that some solution of (2.1) satisfies dQ uE .0/ ıX as well as E.E u/ < J.Q/ C "2 where " ıX . Further assume that the trajectory uE .t/ is not trapped by the 2"-balls around .˙Q; 0/ relative to the dQ -metric. Then uE is ejected from the ıX -ball.
158
4 Above the ground state energy II: Moving away from Q
Proof. If the trajectory does not enter the 2"-ball, then by Lemma 4.5 for some positive time dQ uE .t/ needs to attain ıX , say at time t1 > 0. But then at the time t0 2 Œ0; t1 at which dQ uE .t/ achieves its minimum the ejection condition (4.27) is satisfied and the solution is ejected from the ıX -ball in the sense of the previous definition. Note that (4.26) holds at t D t0 since dQ uE .t0 / > 2". Also, we have t1 t0 1 due to the fact that dQ uE .0/ ıX . In particular, we may wait a constant amount of time until the term CdQ uE .t0 / in the final estimate of (4.28) becomes negligible compared to dQ uE .t/ . This insures that the final estimate of (4.31) is satisfied. On the other hand, assume that uE .t/ does enter the 2"-ball but is not trapped by it. Then it exits this ball again, and the ejection lemma applies starting from that exit time. This again ensures that the solution is ejected as claimed.
4.2
J and K0 ; K2 above the ground state energy
4.2.1 The variational estimates The following lemma implies that sign of both K0 and K2 cannot change outside of a neighborhood of .Q; 0/ that is not too small – the size here depends on the amount by which the energy of the solution exceeds that of Q. It is essential to note that the following lemma is noneffective, i.e., the dependence of "0 .ı/ on ı is nonexplicit. Therefore, one can only use Lemma 4.8 with a fixed constant ı > 0. Lemma 4.8. For any ı > 0, there exist "0 .ı/; 0 ; 1 .ı/ > 0 such that for any uE 2 H satisfying E.E u/ < J.Q/ C "20 .ı/;
dQ .E u/ ı;
(4.32)
one has either K0 .u/
1 .ı/
and
K2 .u/
1 .ı/;
(4.33)
or 2 K0 .u/ min 1 .ı/; 0 kukH 1
2 and K2 .u/ min 1 .ı/; 0 krukL 2 : (4.34)
Proof. 0 is an absolute constant that will be determined via the constant in (2.26). First we prove the conclusion separately for s D 0 and s D 2 by contradiction. Fix
159
4.2 J and K0 ; K2 above the ground state energy KD0
E WD E.u; u t / > J.Q/ C "2 DW J
KD0
K 0 such that ı ıS ;
ı ıX ;
" "0 .ı /;
" R min ı ; 1 .ı /1=2 ; 01=2 ; J.Q/1=2 ; < 0 .M /;
1=6 J.Q/1=2 :
(4.38) (4.39)
Suppose that a solution u.t/ on the maximal existence interval I R satisfies for some " 2 .0; " , R 2 .2"; R ; and 1 < 2 < 3 2 I , E.E u/ < J.Q/ C "2 ; dQ uE .1 / < R < dQ uE .2 / > R > dQ uE .3 / : Then there exist T1 2 .1 ; 2 / and T2 2 .2 ; 3 / such that dQ uE .T1 / D R D dQ uE .T2 / dQ uE .t/ ; .T1 < t < T2 /:
1
Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2008).
165
4.3 The one-pass theorem T2
T1 Figure 4.6. The cutoff w.t; x/ in the one-pass theorem
Lemma 4.9 gives us a fixed sign f˙1g 3 s WD S u.t/ ;
.T1 < t < T2 /:
Now we derive the localized virial identity with a precise error bound. The cutoff function is defined by ( x=.t T1 C S/ t < .T1 C T2 /=2 ; w.t; x/ D x=.T2 t C S/ t > .T1 C T2 /=2 ; where S 1 is a constant to be determined later, and is a radial smooth function on R3 satisfying .x/ D 1 for jxj 1 and .x/ D 0 for jxj 2. Using the equation we have ˝ ˛ 1 Vw .t/ WD wu t j .xr C rx/u ; VPw .t/ D K2 u.t/ C O Eext .t/ ; (4.40) 2 where Eext .t/ denotes the exterior free energy defined by Z Eext .t/ WD e 0 .u/ dx; e 0 .u/ WD juj P 2 C jruj2 C juj2 =2; X.t/ ( 2 jxj > t T1 C S .T1 < t < T1 CT /; 2 x 2 X.t/ ” T1 CT2 jxj > T2 t C S . 2 < t < T2 /: We infer from the finite propagation speed that max Eext .Tj / 1 H)
j D1;2
sup T1 t T2
Eext .t/ . max Eext .Tj / : j D1;2
(4.41)
166
4 Above the ground state energy II: Moving away from Q
To see this, construct global solution vj such that vj D u in X.t/ for jt Tj j < jT1 T2 j=2, and kE v .Tj /k2H . Eext .Tj /, by cutting off the initial data at t D Tj and using the small data theory. The exterior energy at t D Tj is bounded by
2 Eext .Tj / . e 2S C
.T E j / E ; where the term e 2S is dominating the tails of Q and , due to their exponential decay. Hence, choosing S j log Rj 1; we obtain Eext .t/ . R2 , and thus VPw .t/ D
K2 u.t/ C O.R2 /;
.T1 < t < T2 / :
(4.42)
We turn to the leading term K2 . In order to apply the ejection Lemma 4.3, we need the exiting property of the solution (4.27). For that purpose, take any tm 2 ŒT1 ; T2 where dQ uE .t/ attains a minimum in t such that .R / Rm WD dQ uE .tm / D inf dQ uE .t/ < ı : jt tm j tm , decreasS ing for t < tm , and equals ıX on @Im . For each t 2 I 0 n m Im and s D 0; 2, one has .t 1; t C 1/ I 0 , dQ uE .t/ ı and Z
tC1 t 1
min sKs u.t 0 / dt 0 R2 :
sD0;2
(4.46)
By the monotonicity, we can keep applying the above theorem at each t > 2 until dQ .E u/ reaches R . Besides, one concludes that at any later time tm > 2 necessarily E after it is dQ .E u/ > R . In other words, u cannot return to the distance R to ˙Q, ejected to the distance ıX > R .
169
4.3 The one-pass theorem
4.3.3 Vanishing kinetic energy leads to scattering As observed before, in the region K2 0 of Lemma 4.8, K2 .u.t// can vanish at some time if and only if kru.t/kL2x does so, see Lemma 4.8. We now address the ineffectiveness of the lower bound in (2.65). The argument which we used earlier for this purpose, see (2.66)–(2.68), does not apply since it required the energy to lie below that of Q. As clearly demonstrated by the proof of the one-pass theorem, we should treat this kind of vanishing only in the time averaged sense over a sufficiently large interval. It is not hard to see that this leads to global existence and scattering. The idea is that all frequencies have to shift to 0, which leads to the scattering in both time directions by the small Strichartz norm in the regime of subcritical regularity. Technically speaking, one can approximate u by a linear KG-wave v over t 2 Œ0; 2. It then follows that v has small HP 1 norm, which implies that it has small L3t L6x norm globally in time by the subcriticality of that norm. Consequently, one can then approximate u globally in time by v. Lemma 4.13. For any M > 0, there exists 0 .M / > 0 with the following property. Let u.t/ be a finite energy solution of NLKG (2.1) on Œ0; 2 satisfying 2
Z M; kE ukL1 t ..0;2IH//
0
ru.t/ 2 2 dt 2 L
(4.47)
for some 2 .0; 0 . Then u extends to a global solution and scatters to 0 as t ! ˙1, and moreover ku.t/kL3 L6x .RR3 / 1=6 . t
Proof. First we see that u can be approximated by the free solution v.t/ WD e ihrit vC C e
ihrit
v ;
v˙ WD u.0/ ihri
1
u.0/ P =2 :
This follows simply from the Duhamel formula kv
3 ukL1 Hx1 .0;2/ . ku3 kL1 L2x .0;2/ . kukL 3 L6 t
t
.
t
x
2 krukL 2 L2 .0;2/ krukL1 L2 .0;2/ x t t x
2 M ;
if 0 M 1, where we used Hölder’s inequality and the Sobolev embedding HP 1
170
4 Above the ground state energy II: Moving away from Q
L6 . In particular, Z 2
2
rv.t/ 2 2 dt 4 Lx 0 Z h ˚ D C jj2 2jb v C j2 C 2jb v j2 C Im hi
1
.e 4i hi
v 1/b v Cb
i
d
2 2 & krvC kL 2 C krv kL2 ;
where b v denotes the Fourier transform in x of v. Now we use the Strichartz estimate for the free Klein–Gordon equation, see Section 2.5, ke ˙ihrit 'kL3 B 4=9 t
18=5;2 .RR
3/
. k'kHx1 ;
(4.48)
s where Bp;q denotes the Besov space with s regularity on Lp . Combining it with Sobolev, we obtain
kvkL3 L6x .RR3 / . kvkL3 BP 1=3 .RR3 / t t 18=5;2 X . kv˙ kHP 1=3 \HP 8=9 . M 2=3 1=3 C M 1=9 8=9 1=6 ; ˙
if 0 M 4 1. Therefore, we can identify u as the fixed point for the iteration in the global Strichartz norm kukL1 Hx1 .RR3 / . M; t
kukL3 L6x .RR3 / 1=6 ; t
which automatically scatters.
4.4
Summary and conclusion
In the following chapter, we assemble various pieces such as the ejection lemma, the existence of the sign function away from the ground states, and the one-pass theorem into a description of the global dynamics of solutions with energies in ŒJ.Q/; J.Q/C "/. The interval of energies 1; J.Q/ was treated in Chapter 2. The main objective will be to show that the sign of both K0 u.t/ and K2 u.t/ eventually stabilizes. What this means is that for any solution u of the NLKG equation defined on a maximal interval Œ0; T / there exists T 0 2 .0; T / so that these signs are constant and equal for all T 0 < t < T . This is achieved by combining Lemmas 4.8 and 4.3 with the one-pass theorem. Indeed, Lemma 4.8 shows under the energy assumption E.E u/ < J.Q/ C "2 that these signs can only change by making a pass through the
4.4 Summary and conclusion
171
ı."/-balls. The solution trajectories may make no pass through such a ball, which then fixes the signs from the start, or they may also “get stuck” in the ı-ball. But then, due to Corollary 4.7, they then need to be trapped in a 2"-ball (and thus lie on the center-stable manifold). However, in the non-trapping situation, the ejection lemma, Lemma 4.3, guarantees that the solution must exit the ball, and then cannot return due to the one-pass theorem. The constant signs then need to be translated into finite time blowup or scattering to zero, respectively. While the former is a fairly straightforward generalization of the Payne–Sattinger convexity argument from Chapter 2, the latter is more subtle and will be based on the Kenig–Merle argument, as presented in Chapter 2. However, one cannot proceed exactly as in that section and we will need to invoke the one-pass theorem in order to carry out the concentration-compactness argument correctly.
5
Above the ground state energy III: Global NLKG dynamics
In this chapter we combine the results of the previous two chapters to characterize all possible types of long-time asymptotic behavior of the solutions to the cubic NLKG equation in radial R3 . In our first theorem, see Theorem 5.1 below, we ignore the center-stable manifolds and prove a result more in the style of an orbital stability result. That is, instead1 of “scattering to the ground states” one has “trapped by the ground states”. This formulation seems quite natural, as it clearly separates different dynamical issues. To be more specific, once Theorem 5.1 has been proved, the trapping scenario (in forward time) can then immediately be associated with the center-stable manifold due to the repulsivity property of that manifold, see Chapter 3. Furthermore, the dispersive properties of the linearized NLKG flow are needed only to show that trapped solutions scatter to the ground states, but not for the existence of the manifolds themselves (this refers to the distinction between the Hadamard approach on the one hand, and the Lyapunov–Perron method on the other hand). Recall that the linearized NLKG flow refers to the equation vR C LC v D F and establishing its dispersive properties involves some finer spectral properties of LC , namely the issue of eigenvalues in the gap .0; 1/ and the question of a threshold singularity of the resolvent, see (3.12). All of this is then needed when we invoke Theorem 3.22, see Theorem 5.2 and 5.3 below. In Section 5.1 we state the main results on global dynamics. First the 9-set theorem which says that all possible combinations of the forward and backward trichotomies are allowed. We repeat that this does not depend on Chapter 3, and thus also does not invoke the gap property. However, the proof of the scattering statement in Theorem 5.2 does rely on that property and provide a further characterization of those solutions which are trapped by the ground states. Theorem 5.3 characterizes the solutions found by Duyckaerts and Merle as stable manifolds associated with the ground states. 1
In the orbitally stable case, the distinction orbital vs. asymptotic stability can be seen from [67], [143], [144] on the one hand, and [22], [23], [128] on the other hand.
174
5 Above the ground state energy III: Global NLKG dynamics
In Section 5.2 we give the details of the Payne–Sattinger blowup argument for the negative K case, as well as of the scattering to zero for solutions which settle into the K 0 case. The “settling in” here refers to the fact that the one-pass theorem of the previous chapter guarantees that the sign of K stabilizes for any solution. More precisely, we shall show that if I is the open maximal time interval of existence, then we can remove a closed subinterval from I such that in the remaining open subintervals of I the signs of Kj .u.t// is fixed. The scattering argument is quite delicate, and is based on the Kenig–Merle approach; we emphasize that in contrast to the original form of the Kenig–Merle argument, the one-pass theorem plays an important role here, too. In the final Section 5.3 we put everything together to obtain the global dynamical picture as described by the theorems of Section 5.1.
5.1
Statement of the main results on global dynamics
Following [109], we will prove several results about the global behavior of solutions to (2.1) with energy at most slightly above that of the ground state E C "2 g: H" WD fE u 2 H j E.E u/ < E.Q/ Note that the only symmetry in H is u 7!
(5.1)
u in this setting.
Theorem 5.1. Consider all solutions of NLKG (2.1) with radial initial data uE .0/ 2 H" for some small " > 0. The set of all these solutions splits into nine non-empty sets characterized as (1) Scattering to 0 for both t ! ˙1, (2) Finite time blow-up on both sides ˙t > 0, (3) Scattering to 0 as t ! 1 and finite time blow-up in t < 0, (4) Finite time blow-up in t > 0 and scattering to 0 as t ! 1, (5) Trapped by ˙Q for t ! 1 and scattering to 0 as t ! 1, (6) Scattering to 0 as t ! 1 and trapped by ˙Q as t ! 1, (7) Trapped by ˙Q for t ! 1 and finite time blow-up in t < 0, (8) Finite time blow-up in t > 0 and trapped by ˙Q as t ! 1, (9) Trapped by ˙Q as t ! ˙1, where “trapped by ˙Q” means that the solution stays in a O."/ neighborhood of E forever after some time (or before some time). The initial data sets for (1)–(4), ˙Q respectively, are open.
175
5.1 Statement of the main results on global dynamics W cu
I III VI
IX D W c V
VII
.Q; 0/
W cs
IV VIII
II
Figure 5.1. Illustration of the nine sets
In contrast to the Payne–Sattinger result, here one has solutions which blow up for t < 0 and scatter for t ! C1, or vice versa. It also implies that the initial data set for the forward scattering .1/ [ .3/ [ .6/ (or backward scattering) is unbounded in H; in fact, it contains a curve connecting zero to infinity in H. The number “nine” simply means that all possible combinations of scattering to zero/scattering to ˙Q/finite time blow-up are allowed as t ! ˙1. Each of these are in fact realized by infinitely many solutions. Due to the Bates–Jones theorem present in Chapter 3 one can characterize the trapping property in terms of the invariant manifolds associated with the ground state, see in particular Corollaries 3.16 and 3.18. For this, the spectral information provided by Lemma 3.2 is sufficient. While the Bates–Jones theorem only applies to powers p 3 in the nonlinearity (in R3 ), it can be extended to cover the full energy subcritical range in all dimensions, see [112]. Thus, the trapping alternative in Theorem 5.1 can be characterized as follows: any solution which is trapped by Q as jtj ! 1 necessarily belongs to the center manifold W c , and conversely, whereas those trapped as t ! C1 (resp. 1) belong to W cs (resp. W cu ). In particular, we see that .9/ is a co-dimension 2
176
5 Above the ground state energy III: Global NLKG dynamics
Lipschitz graph, whereas .5/ is co-dimension 1, etc. To summarize: Theorem 5.2. The sets .5/ [ .7/ [ .9/ and .6/ [ .8/ [ .9/ are codimension one Lipschitz manifolds in the (radial) phase space H, and they are the center-stable E Similarly, (9) is manifold, respectively the center-unstable manifold, around ˙Q. a Lipschitz manifold of codimension 2, namely the center manifold. Using the gap property (3.12) of LC , one can show that all solutions on the center-stable manifold E D .˙Q; 0/ in forward time, and similarly for the center-unstable and scatter to ˙Q center manifolds. The regularity can be improved but we do not pursue this here (in fact, Proposition 3.22 constructs smooth manifolds, but the graph transform approach does not). Recall from [44] that (3.12) fails if one lowers the power 3 in the nonlinearity slightly, say to power < 2:8. Thus, the final scattering statement of the previous theorem is somewhat delicate. On the other hand, our argument for Theorem 5.1 as well as for the non-scattering part of Theorem 5.2 are quite robust. Furthermore, it is straightforward to extend Theorem 5.1 to all L2 super-critical and H 1 subcritical powers and all space dimensions, i.e. uR
u C u D up ;
1 C 4=d < p < 1 C 4=.d
2/;
u W R1Cd ! R:
Finally, from the above theorems, we can deduce a Duyckaerts–Merle type result at the energy threshold, cf. [48]. Theorem 5.3. Consider the limiting case " ! 0 in Theorem 5.1, i.e., all the radial E Then the sets (3) and (4) vanish, while the sets solutions satisfying E.E u/ E.Q/. (5)–(9) are characterized, with some special solutions W˙ , as follows: .5/ D f˙WE .t
t0 / j t0 2 Rg;
.6/ D f˙J WE . t
t0 / j t0 2 Rg;
.7/ D f˙WEC .t
t0 / j t0 2 Rg;
.8/ D f˙J WEC . t
t0 / j t0 2 Rg;
E .9/ D f˙Q.t
t0 / j t0 2 Rg
with J being the standard symplectic 2 2-matrix. The solutions WE˙ .t/ converge E as t ! 1, in fact at the rate e jkjt where k 2 is the lowest exponentially to Q eigenvalue of LC . As mentioned before .5/[.7/[.9/ is the stable manifold, and .6/[.8/[.9/ the E The remainder of this chapter consists of unstable manifold, associated with ˙Q. the proofs of these three results. While most of the technical work has already been
5.2 The blowup/scattering dichotomy in the ejection case
177
done, we still need to provide the proofs of the blowup/scattering to zero dichotomy for solutions which are ejected away from the ground states.
5.2
The blowup/scattering dichotomy in the ejection case
5.2.1 Blowup after ejection Here we prove that the solution u with S uE .2 / D 1 in Theorem 4.11 blows up in finite time after time 2 . This will be done by means of the contradiction argument of Payne–Sattinger, which relies on the functional K0 . Thus, suppose that u extends to all t > 2 and let
2 y.t/ WD u.t/ L2 : From the NLKG equation (2.1) we have 2 : yR D 2 kuk P L 2 C sK0 u.t/ x
(5.2)
Applying the lower bound on K0 in Theorem 4.11 to the integral yields Z X Œy P1 & ı C R2 dt D 1 ; X 2 Im
I0
and so y.t/ ! 1 as t ! 1. Then from (5.2), yR
2 2 2 8E.E u/ C 6kuk P L P L P 2 =.2y/; 2 C 2kukH 1 6kuk 2 3y
for large t, where we used Cauchy–Schwarz for yP D 2hujui. P Hence, @2t .y
1=2
/D
.2y 3=2 /
1
Œy yR
3yP 2 =2 0 ;
which contradicts that y ! 1 as t ! 1. Therefore, u does not extend to t ! 1.
5.2.2 Scattering after ejection For the solution u with S uE .2 / D C1 in Theorem 4.11, the forward global existence follows from the energy bound of Lemma 4.10. In analogy with Chapter 2,
178
5 Above the ground state energy III: Global NLKG dynamics
we prove the scattering to 0 for t ! 1 by the contradiction argument of Kenig– Merle. Thus, we use the functional K2 and the virial identity. While the argument is of course similar to the one in Chapter 2, it is also quite different in so far as we crucially depend on the one-pass theorem, see Theorem 4.11. The point here is that the profile decomposition of the concentration-compactness argument needs to be performed subject to constraints, cf. (5.3) below. The induction on energy step then requires that the building blocks in the profile decomposition respect these constraints. We urge the reader to go through the scattering argument for the radial case in Chapter 2 before reading this section. In particular, we shall freely use the perturbation lemma, see Lemma 2.19, as well as the profile decomposition given by Proposition 2.17 of that chapter. To be more specific, fix " 2 .0; " / and let U."; R / be the collection of all solutions u of NLKG on Œ0; 1/ satisfying E.E u/ J.Q/ C "2 ; dQ uE Œ0; 1/ ŒR ; 1/; S uE Œ0; 1/ D C1: (5.3) Note that the first two conditions imply that uE Œ0; 1/ H.ı / so that we can use Lemma 4.9 to define S.E u/. By the remark following Theorem 4.11, any solution with S D C1 in that theorem will eventually satisfy the above conditions. For each E > 0, let M.E/ be a uniform Strichartz bound defined by ˚ u/ E ; M.E/ WD sup kukL3 L6x .0;1/ j u 2 U."; R /; E.E (5.4) t where we chose the norm L3t L6x to be an H 1 subcritical and non-sharp admissible Strichartz norm such that its finiteness implies scattering. We know from Chapter 2 that M.E/ < 1 for E < J.Q/. In fact, in that case a uniform bound holds globally in time, i.e., for L3t L6x .R/. In order to extend this property to J.Q/ C "2 , put ˚ E ? D sup E > 0 j M.E/ < 1 (5.5) and assume towards a contradiction that E ? < J.Q/ C "2 : We consider the nonlinear profile decomposition for any sequence uE n 2 U."; R / satisfying as n ! 1 E.E un / ! E ? ;
kun kL3 L6x .0;1/ ! 1: t
(5.6)
We are going to show that the remainder in the decomposition vanishes and there is only one profile which is a critical element, i.e., uE ? 2 U."; R /;
E.E u? / D E ? ;
ku? kL3 L6x .0;1/ D 1: t
(5.7)
5.2 The blowup/scattering dichotomy in the ejection case
179
By the arguments in the scattering part of Chapter 2 it then follows that the forward trajectory of uE is precompact in H. It then follows by essentially the same arguments based on the virial identity (2.64) as in Chapter 2 that such a critical element cannot exist (this is the so-called rigidity step). To commence the detailed argument, we apply the Bahouri–Gérard decomposition, see Proposition 2.17. Before applying the profile decomposition, we translate un in time in order to achieve the following estimates: 2 (5.8) dQ uE n .0/ > ıX ; K2 un .0/ "2 : 3 Since dQ .E un / remains above R , Corollary 4.7 implies that there exists 0 Tn . k 1 log.ıX =R / so that dQ uE n .Tn / ıX . Since S D C1, Lemma 4.10 implies that kun kL1 H.0;1/ M . Since kun kL3 L6x ! 1, by the same argument t as for (4.45) we deduce that there exists 0 Tn0 near Tn such that 2 dQ uE n .Tn0 / > ıX : 3 0 Translating un WD un .t Tn /, we obtain (5.8) in addition to (5.6). Now apply Proposition 2.17 to the free solution with the same initial data as un . Using the notation of that proposition, we set vnj D v. C tnj / and let wnj be the nonlinear solution with the same data as vnj at t D 0. In other words, X U.t/E un .0/ D vEnj C Enk ; w E nj .t/ D U N .t/E vnj .0/; K2 un .Tn0 / R2 > 2"2 ;
j t1 . Then 0 0 0 t/ is . If u0 does not scatter as t ! 1, then u0 .t1 dQ uE .t/ R for t t1 0 a critical element. Suppose that u scatters as t ! 1. Then the profile decomposition is a good approximation of un on . 1; t tn0 / for large n. Therefore, by the perturbation lemma for large k and n, 0 " & dQ uE 0 .t / C O."/ lim sup dQ uE n .t t1 / R ; n!1
which contradicts " R . In conclusion, for some T 2 R and s 2 f˙1g, we have shown that u0 .st C T / is a critical element. Since un is a minimizing sequence, it implies also that the other components must vanish strongly in the linear profile decomposition. Therefore, along a subsequence,
lim uE n .Tn0 / uE 0 .tn0 / H D 0 ; n!1
where Tn0 0 is the time shift for (5.8). Both Tn0 and tn0 are bounded from above as n ! 1. If tn0 ! 1, then u0 scatters for t ! 1, and so kun kL3 L6x . 1;Tn0 / is t bounded for large n, by the local theory of the wave operator. Applying this to the sequence of solutions un WD u0 .t C n / for arbitrary n ! 1, one obtains the precompactness of the forward trajectory of the critical element,
182
5 Above the ground state energy III: Global NLKG dynamics
and a contradiction from a localized (time-independent) virial identity together with the lower bound on K2 . These steps are essentially the same as in the radial scattering proof in Chapter 2. Thus we conclude that no solution u satisfies (5.7), and therefore E ? D J.Q/ C "2 . We summarize our findings in the following proposition. Proposition 5.4. For each " 2 .0; " , there exists 0 < M J.Q/ C "2 < 1 such that if a solution u of NLKG (2.1) on Œ0; 1/ satisfies E.E u/ J.Q/ C "2 , dQ uE .t/ R and S uE .t/ D C1 for all t 0, then u scatters to 0 as t ! 1 and kukL3 L6x .0;1/ M . t
Note that the uniform Strichartz bound can be valid only on the time interval E by a fixed distance. Indeed, without where the solution is already away from ˙Q this separation one cannot hope for any uniform bound, even for those solutions scattering for both t ! ˙1. This is due to the fact that the solutions can stay close E for arbitrarily long time. to ˙Q Exercise 5.5. Prove the claim made in the previous sentence.
5.3
Proofs of the main results
5.3.1 Proofs of Theorems 5.1, 5.2 Fix 0 < " " and let H" WD fE u 2 H j E.E u/ < J.Q/ C "2 g be the initial radial data set. We can define the following subsets according to the global behavior of the solution u.t/ to the NLKG equation: for D ˙ respectively, ˚ S" D uE .0/ 2 H" j u.t/ scatters as t ! 1 ; ˚ T" D uE .0/ 2 H" j u.t/ trapped by f˙Qg for t ! 1 ; (5.12) ˚ " " B D uE .0/ 2 H j u.t/ blows up in t > 0 : The trapping for TC" can be characterized as follows, see Corollary 4.7: there exists T > 0 such that for all t T one has dQ uE .t/ 2": Obviously those sets are increasing in ", and have the conjugation property ˚ " " X D u.0/; u.0/ P 2 H" j uE .0/ 2 X˙ ;
5.3 Proofs of the main results
183
for X D S; T ; B. Moreover, SC and TC are forward invariant by the flow of NLKG, while S and T are backward invariant. We have proven in the previous sections that " H " D SC [ TC" [ B"C D S" [ T " [ B" ; " the disjoint union for each sign. It follows from the scattering theory that S˙ are " open. We claim the same for B˙ , which is not a general fact. Pick any solution uE .t/ in B"˙ which blows up at some finite time T , say. Then kE u.t/kH ! 1 as t ! T . Next, one has @ t ku.t/k22 D 2hujui, P as well as 2 2 2 2 @2t kukL P L K0 u.t/ 6kuk P L 8E.E u/ : 2 D 2 kuk 2 2 C 2kukH 1 x x x x From the lower bound on the right-hand side we conclude that K0 u.t/ ! 1 as t ! T 0. This means that the slope of hu.t/ju.t/i P ! 1 as t ! T 0, whence also hu.t/ju.t/i P ! 1 as t ! T 0. We claim that if T < T is very close to blow-up time, then every solution starting in B1 uE .T / , the unit-ball around uE .T / relative to H, necessarily blows up in the positive time direction. By the Payne–Sattinger argument, we only need to show that for any such solution K0 .u.t// < 0 as long as it is defined. It is clear that this condition will hold initially. By Lemma 4.8 with the choice of " in (4.38), we further see that it can be E of size ıS . However, violated only if the solution returns to a neighborhood of ˙Q in that case necessarily jhujuij P . 1, which is impossible since hujui P starts off very large and has to increase as long as K0 u.t/ < 0. Thus, any solution close to u E and has to blow up sooner or later, as around t D T cannot come close to ˙Q " claimed. Therefore, B˙ are also open, so T˙" are relatively closed in H" . " Since B"C and SC are disjoint open sets, they are separated by TC" . I.e., any two " " points from SC and BC cannot be joined by a curve without passing through TC" . We now begin with the construction of solutions that exhibit any one of the 9 E it is easy to see by different scenarios described above. In a small ball around ˙Q, " " means of the linearized flow that the open intersections B˙ \ S , B"C \ B" and " " SC \ S are all non-empty for any " > 0. In fact, in order to obtain a solution uE in B"C \ S" , we choose initial data such that
.0/ D 0;
P .0/ D k";
.0/ E D 0;
(5.13)
for some 0 < 1. Then the energy constraint E.E u/ < J.Q/ C "2 is satisfied, and by the same argument as in Lemma 4.3, we have j.t/
0 .t/j . e 2kjt j 2 "2 ;
k .t/kE . " C e 2kjtj 2 "2 ;
184
5 Above the ground state energy III: Global NLKG dynamics
as long as e kjtj " ıX , where the free solution 0 is given in this case 0 .t/ D sinh.kt/" : E both in ˙t > 0. Since .t/t > 0 for Hence, uE .t/ exits the R neighborhood of ˙Q kjtj " e " ıX , we obtain S uE .t/ t < 0 after exiting, and so u blows up in " t > 0 and scatters for t ! 1. Therefore, B"˙ \ S are both non-empty. " " \ S" , respectively. In the same way, we construct solutions in BC \ B" and SC More precisely, let u.˙/ be those solutions for which is approximated by the free solutions .˙/ 0 .t/ D ˙ cosh.kt/" ; " respectively. Then uE .C/ 2 B"C \ B" and uE . / 2 SC \ S" by the same considerations as before. Let uE 2 B"C \ S" be the solution with initial data (5.13). Then for any " e k t " "1=2 one has the distance estimates
uE .t/ uE .C/ .t/ . e 2kt 2 "2 C " " ; dQ uE .t/ & e kt " " : E
Hence, there exists a curve H" joining uE .t/ and uE .C/ .t/ within the region S.E u/ < 0 and dQ .E u/ ". Since uE .t/ 2 S" and uE .C/ .t/ 2 B" , there exists a point E as p0 2 T" \ . Since the solution starting from p0 enters the 3"-ball around ˙Q t ! 1, and initially p0 is much further away and S.p0 / < 0, we conclude by the one-pass theorem that p0 2 B"C . Hence, T " \ B"C is non-empty as well. In the same way, we can find a point on the curve connecting uE .t/ and uE . / .t/ for some t < 0, " which is in TC" \ S" . Therefore, T˙" \ B" and T˙" \ S are both not empty. Taking the limit " ! C0, it is easy to observe that they contain infinitely many points on different energy levels. E is of course nonempty, but there are infinitely many Finally, TC" \ T " 3 Q points besides Q, which can be seen by restricting H" to the hyperplane uP D 0 (which guarantees that the resulting solutions are even in time): ˚ H"uD0 WD .u; 0/ j J.u/ < J.Q/ C "2 3 u.˙/ .0/ : P . / Since u.C/ .0/ 2 BC .0/ 2 S"C , and there are infinitely many connecting " and u " curves in HuD0 , there are infinitely many points in TC" \ H"uD0 separating B"C and P P " SC . The symmetry u.t/ 7! u. t/ implies that
TC" \ H"uD0 TC" \ T " : P
5.3 Proofs of the main results
185
Thus we have shown that the nine solution sets are all non-empty. " By using scattering theory, we can prove that SC and the fixed energy section N (with U being the NLKG evolution) ˚ D" SC WD uE 2 H j U N .t/E u scatters as t ! 1 ; E.E u/ D J.Q/ C "2 " D" are both pathwise connected. To see this, let uEj 2 SC or SC for j D 0; 1 and let uEj C be their asymptotic profiles
N
U .t/E uj U.t/E uj C E ! 0; .t ! 1/ :
There exists a continuous curve uE C W Œ0; 1 ! H such that kE uC k2H D .1
/kE u0C k2H C kE u1C k2H ;
as well as u solving NLKG on some .T ; 1/ such that
uE .t/ U.t/E uC E ! 0; .t ! 1/ : Moreover, T and uE .T C 1/ are also continuous in . The scattering property implies that uE .t/ 2 SC for any t > T , and E.E u / D kE u C k2H =2 D .1
/E.E u0 / C E.E u1 / :
Let T WD sup01 T C 1 < 1, then uEj and uEj .T / are connected by the flow, " D" while uE 0 .T / and uE 1 .T / are connected by uE .T /, all included in SC or SC . This proves the connectedness of those sets. Next, we observe that TC" contains the center-stable manifold for the t 0 direction, as constructed in Theorem 3.22. In fact, by the uniqueness property of that manifold, TC" is identical to the maximal backward extension of all solutions on this manifold. Since the flow is C 1 in both directions, T˙" are indeed connected, 1-codimensional and smooth manifolds in the energy space. Moreover, a sufficiently " small ball B around each point of TC" is separated by the hypersurface TC" into SC \B " and BC \ B. P Reversing time, the unThe linear approximation is the hyperplane k D . stable and stable directions are interchanged. Heuristically speaking, but still in a precise sense, the classification into those nine sets appears as the symbol ˝ in the P phase plain around Q, see Figure 5.1 (which also appears on the cover). The .; / meaning of this is as follows: the four chambers correspond to all possible combinations of finite time blowup vs. scattering to zero as t ! ˙1. The boundaries of
186
5 Above the ground state energy III: Global NLKG dynamics
these chambers, more precisely, the four straight line segments in ˝, correspond to the four possible combinations of scattering to Q as t ! ˙I on the one hand, and finite time blowup/scattering to zero as t ! 1 on the other hand. Finally, the dot in the middle corresponds to those solutions which are trapped by .˙Q; 0/ as t ! ˙1. In the full infinite dimensional representation, the aforementioned “dot” is a smooth manifold of codimension 2, and is given as the intersection of the two center-stable manifolds corresponding to the forward and backward evolutions, respectively; the latter being of course the center-unstable manifold. The reader should bear in mind that reversing the sign of the time is tantamount to replacing the data .u0 ; u1 / at time t D 0 with .u0 ; u1 /. We also remark that there can be no solution which is trapped by Q as t ! 1, and Q as t ! 1, as this would contradict the one-pass theorem. So we conclude that the only possible scenario is trapped by either Q or Q in both time directions. In the geometric pictures, this corresponds to the two center manifolds E and Q, E respectively. We remark that Bates and Jones already established around Q the orbital stability globally in time for solutions which lie on these manifolds (at least for powers not exceeding 3), see [6]. To be more precise, we now prove that TC" consists of two connected components " TC.˙/ defined by ˚ " TC./ D uE 2 H" j U N .t/E u is trapped by Q as t ! 1 : " To see that TC.C/ is pathwise connected, take any pair of points from that set and send them by the nonlinear flow until they land on the local center-stable manifold, which is of course connected. " " To see the separation of TC.C/ from TC. , suppose for contradiction that p is / a common point of their boundaries. Since these manifolds are relatively closed in H" , it follows that p 2 TC2" . Thus, U N .t/p will be trapped by Q with D ˙. E But Then its small neighborhood can be mapped by the flow into a ball around Q. " " it contains a point in TC. / , contradicting the one-pass theorem. Thus TC consists " of two connected components TC.˙/ . We can also observe that " D" @SC D TC" [ TCD" [ SC
is connected. The inclusion follows from the fact that B"C and fE.E u/ > J.Q/ C "2 g are open sets, whereas is easily seen by perturbing the data from the right. The connectedness follows from that of the following three sets < D" TC.C/ [ TC.C/ ;
D" SC [ TCD" ;
< TC.
/
D" [ TC. /:
187
5.3 Proofs of the main results
The Payne–Sattinger solutions can be classified in our notation, see (5.12), by 0 SC D S0 ;
B0C D B0 ;
TC0 D T 0 D ; :
This statement follows from our theorems, because the energy constraint E.E u/ < E prohibits the sign change of Ks and scattering to ˙Q. The threshold case E.Q/ given by Theorem 5.3, i.e., the analogue of the Duyckaerts–Merle theory [48], requires a little more work. Here we give a proof using the scattering to the ground state, hence the spectral gap property, but it is not essential. See [112] for a proof in the general case using the graph transform method.
5.3.2 Proof of Theorem 5.3
e
0 Let X˙ D
T
">0
" X˙ for X D S; B; T . Then Theorem 5.3 can be restated as
e
f0 f 0 0 0 Sf C \ B D S \ BC D ;;
f0 0 E Tf C \ T D f˙Q.t
e
s/ j s 2 Rg;
f0 0 E Sf s/ j s 2 Rg; B0C \ Tf0 D f˙WEC .t s/ j s 2 Rg; C \ T D f˙W .t f f0 \ Tf0 D f˙J WE .s t/ j s 2 Rg; 0 0 E S \ Tf t/ j s 2 Rg; B C C D f˙J W .s C f f0 f0 0 0 for some solutions W˙ . To see that Sf C \ B D ;, let u be a solution in SC \ B . Theorem 4.11 implies that there exists a finite interval I" R for any " > 0, E D I" . Then I" is decreasing as " ! C0, and hence by such that uE 1 .B3" .˙Q// E which implies u Q, a continuity of uE , there exists t 2 R such that uE .t/ D Q, contradiction. 0 Next, any solution u in Tf C corresponds to the case 1 0 in Theorem 3.22, and it is therefore parametrized by .0/ 2 R. Since .t/ ! 0 as t ! 1, the uniqueness in the proposition implies that there are only three solutions converging E modulo time translation: one with > 0 decreasing (for large t), another with to Q, < 0 increasing, and yet another with 0, i.e., Q. Let W˙ be the two solutions and 1 0. Since E.WE˙ / D J.Q/ and W˙ 6D Q, the sign given by .0/ D ˙ 10 of K0 is fixed on their trajectories. Indeed, Lemma 4.3 applies to them backward from a neighborhood of t D 1, from which we deduce that S.WE˙ .t// D 1 away f0 and W 2 S f 0 from t D 1, and so WC 2 B . To prove that W˙ approach Q exponentially, we return to (3.59) and (3.68) with 1 D 0, estimating k.; /kX k WD ke k min.t T
T;0/
.t/kL1 \L1 .0;1/ C ke k min.t
T;0/
.t/kSt.0;1/
188
5 Above the ground state energy III: Global NLKG dynamics
for T ! 1. The same argument as in the proof of Proposition 3.22 yields
.; / k . e kT j.0/j C .; / 2 C .; / .; / k XT X X XT
. e kT C .; / X k ; T
and so .; /
k XT
5.4
.e
kT
, as desired.
Summary and conclusion
This chapter presents the main results for the radial cubic NLKG equation in R3 obtained in this monograph. They are based on the authors’ paper [109]. We now try to summarize the main steps of the method: ı The basic well-posedness theory, and global existence and scattering for small data. This was done in Chapter 2, and involves both energy and dispersive estimates. Central to the scattering analysis is the Strichartz norm k kL3 L6x . t ı A Payne–Sattinger type theory. This refers to the variational theory of the ground state as well as to its dynamical implications for energies strictly below the ground state energy. Of central importance are the functionals K0 ; K2 derived from the stationary energy J via symmetries (dilations of the dependent and independent variables). Essential to the theory in its present form is the fact that the linearized operator LC associated with the ground state exhibits a single negative eigenvalue. This step includes a proof of the scattering property via the Kenig–Merle concentration-compactness method. ı The perturbative analysis near the ground states. This refers to the hyperbolic dynamics near the ground states and involves the construction of the invariant manifolds associated with it. The aforementioned unique negative eigenvalue generates 1-dimensional stable and unstable manifolds, and the center-manifold is of codimension 2. Generally speaking, the construction of the stable and unstable manifolds is less delicate than that of the center-stable one; furthermore, if one is willing to give up the scattering property typically expected of solutions starting on the center-stable manifold, then the construction of that manifold is also more robust (this distinction refers to the Lyapunov–Perron method on the one hand, and the Hadamard method on the other hand). Of central importance is the ejection lemma which describes a mechanism by which to select trajectoE As they are being ejected, ries which are ejected from a small ball around ˙Q. the sign of the functionals K0 and K2 is determined entirely by the sign of the coefficient of the unstable mode. For the latter, it is important that the trajectory
5.4 Summary and conclusion
189
E whose square is at least as large as the energy starts off from a distance to ˙Q E excess E.E u/ E.Q/. ı A one-pass theorem. This step excludes trajectories which pass through small E more than once. The proof of this step hinges on a suitable virial balls around ˙Q identity in combination with the ejection lemma and the variational structure near the ground state. A crucial role is played by a definite sign that is attached to a E whose radius squared solution as long as it remains outside of a ball around ˙Q is large compared to the aforementioned excess of energy (which therefore needs to be small). ı Description of the global-in-time dynamics. By combining the previous four steps one arrives at the global picture as in Figure 5.1. Additional work is required for the scattering property which is obtained by an adaptation of the Kenig–Merle method to solutions with small energy excess. The one-pass theorem plays an important role in this proof. Duyckaerts–Merle type threshold solutions (referring to solutions on the center-stable manifold whose energy is precisely that of the ground state) appear naturally in our theory as the stable and unstable manifolds. However, in order to reach this conclusion one needs to construct the centerstable manifold by means of the Lyapunov–Perron method which in turn hinges on fine spectral properties of the linearized operator (the “gap property” of the linearized operator). In the following chapter we will describe various other scenarios where the authors, in part jointly with Joachim Krieger, have succeeded in carrying out these steps. For example, for the critical wave equation the second step above is precisely the Kenig– Merle theory for that equation, whereas for subcritical NLS this step was carried out by Holmer and Roudenko.
6
Further developments of the theory
So far, we have mostly limited ourselves to the radial, three-dimensional cubic NLKG equation. This had the advantage of providing a convenient framework in which to develop the ideas around the center manifold and the one-pass theorems. However, several questions now pose themselves naturally: (i) Are there any theorems analogous to those of the previous chapter for the same equation but with non-radial data? (ii) Does the same theory apply to the NLKG equation with other powers and in other dimensions? In particular, what happens in the one-dimensional setting where there is much less dispersion? (iii) Are there any analogous results for other dispersive equations which admit soliton solutions such as the NLS equation (for which any stationary solution is modulated by a finite set of parameters)? (iv) What can be said about the energy critical case? Do center-stable manifolds exist in that setting, and is there a 9-set theorem? In this chapter we present some partial answers to these questions. We remark that this chapter is quite different from the previous ones as it does not strive to give a complete exposition with detailed proofs. In fact, we only intend to provide an introduction to, and summary of, the results of [87], [88], [110], [111]. We hope that the reader will consult these references for further details. As for question (i), a complete answer was found in [111] for the cubic NLKG equation in R3 . In addition to the translation symmetry, the nonradial equation also exhibits Lorentz symmetry. The latter can be used to transform a general solution into one with vanishing momentum (at least if jEj > jP j where E; P are the nonlinear energy and the momentum, respectively). The condition E < J.Q/ C "2 2 now becomes Em < J.Q/ C "2 , where Em D E 2 P 2 , provided the difference here is positive. The energy Em is Lorentz invariant, and is called minimal energy, since it is the true energy if P D 0. With the minimal energy instead of the regular energy, the 9-set theorem carries over to nonradial solutions, albeit with a somewhat different proof. The most significant differences from the radial setting occur in the construction of the center-stable manifold. Here one needs to introduce a wider family of ground states, which reflect the effect of a Lorentz contraction in the direction of motion. As in the radial case the key to the construction of the center stable manifolds lies with a suitable dispersive estimate for the linearized operator. It turns out that here one needs to allow for a gradient term with a small time-dependent
192
6 Further developments of the theory
(but space independent) coefficient in the linearized operator. But by means of a variant of the approach in Beceanu [8] this can be handled as well. Throughout the entire nonradial analysis we use a particular complex formalism which is designed, amongst other things, to keep the modulation equations of the first order. We give a brief account of these developments in Section 6.1 below. As for question (ii), we already noted before that the 9-set theorem carries over to all dimensions d 1 and nonlinearities jujp 1 u in the range 1 C d4 < p < 2 1. As one would expect, it is more delicate to prove the scattering to Q, see Theorem 5.2. At least in terms of our current methodology, this requires understanding the spectral properties of the linearized operator LC (the “gap property”, see (3.12)). In Section 6.2 below we follow [88] to establish the existence of center-stable manifolds for the 1-dimensional NLKG with powers p > 5. The novel aspect here (as compared to the three-dimensional situation) is that the Strichartz estimates only go down to L4t which is insufficient for nonlinearities of the form Qp 2 v 2 . Note that these nonlinearities arise in the perturbative argument in the construction of the center-stable manifolds. In order to control these terms, we establish estimates of the type L2t L2x;loc for v, which are local in space. This is natural, as Q provides the needed localization. Estimates of this type have been used previously, for example by Mizumachi [107]. They hinge on the fact that the linearized operator LC does not have a threshold resonance. The advantage of the one-dimensional equation is the explicit spectral theory of LC . It has been known for some time, see for example [90] for an account of this, that for powers p > 3 there are no eigenvalues in the gap and that the thresholds are not resonances. In other words, the “gap property” is completely understood in the 1-dim case. As for question (iii), [110] obtains the full analogue of the NLKG results for the NLS equation in the R3 for radial data (nonradial data are not considered here). Below we give a brief account of the results in [110]. In contrast to NLKG, we need to control two symmetry parameters, namely scaling and modulations of the phase. While the former is essentially fixed by the conserved mass, one can “mod out” the latter. In fact, this is true for the orbital stability statements as in the 9-set theorem, but the construction of the center-stable manifold requires more work and one has to control the global (in forward time, say) dynamics of these symmetry parameters (as is typical of asymptotic stability theory), see the final section of Chapter 3. For the orbital stability part, the underlying symplectic structure provides a convenient framework in order to set up the modulational equations. As far as the 9-set theorem for NLS is concerned, the corner-stone is the same type of one-pass theorem as for the NLKG equation. The proof of this result is somewhat more subtle in the NLS case. For example, a time-dependent cut-off as in Figure 4.6 has little meaning here due to the absence of finite propagation speed.
6 Further developments of the theory
193
Instead, the authors [110] used a virial-type identity with a time-independent weight which goes back to Ogawa–Tsutsumi [113]. The choice of the weight is the delicate part, since one wishes to obtain almost monotone quantities whose derivatives are “mostly” controlled by the sign of K.u/. We remark that the radial assumption appears to be essential for this virial analysis. On the other hand, the restriction to the cubic power and dimension three can easily be relaxed for the purposes of the 9-set theorem. As already for the NLKG equation, it is really the construction of the center-stable manifold via the Lyapunov, Perron method which is more sensitive to the nonlinearity as well as the dimension. For more on these issues see Chapter 3, especially the final section on NLS. Finally, for question (iv) we only present partial answers. The difficulties appear to be more substantial in the energy critical setting. Unlike the subcritical case, the critical equation does not have a mass term (otherwise there are no ground states), i.e., the equation becomes the energy critical wave equation. These equations admit the special stationary solutions called Aubin–Talenti solutions; they are extremizers for the critical Sobolev imbedding HP 1 .Rd / ,! L2 .Rd / and have a simple explicit form. Due to the absence of the mass term, the equations now also admit a scaling symmetry which generate a radial zero mode of the linearized operator LC . In R3 this mode is a zero energy resonance, whereas in dimensions dim 5, it is a zero energy eigenvalue. For all powers of the nonlinearity, one retains the property that LC has a unique negative eigenvalue. While [91] constructs a center-stable manifold of codimension 1 in a suitable topology1 for the critical wave equation in R3 , we do not address the question of existence of this manifold in the energy topology. However, following [88] we can describe the behavior of radial solutions which are not trapped by a “tube” around the rescaled ground state. Again, a one-pass theorem holds here as well. The key to controlling the scaling parameter during the hyperbolic dynamics near the ground state is the fact that the zero mode of LC evolves according to some algebraic law, whereas the exponentially unstable mode grows much more strongly and therefore dominates everything else. By means of this observation, one obtains a “4-set” theorem describing the dynamics outside of the tube. Of essential importance in our argument is the characterization of type-II blowup by Duyckaerts, Kenig, and Merle [47] (type-II refers to blowup for which the free energy remains uniformly bounded as the time approaches blowup time). Indeed, this replaces the subcritical argument that the Payne–Sattinger set P SC is bounded in H 1 L2 and 1
This topology is much stronger than the energy topology. In particular, it is not invariant under the nonlinear flow. This makes it somewhat of a misnomer to call the manifold in [91] “center-stable”. However, it does exhibit the desired asymptotic stability property for any solution starting on it. In this regard, also see the work of Karageorgis, Strauss [81] on the wave equation with juj5 nonlinearity.
194
6 Further developments of the theory
therefore only admits global solutions. For the energy critical equation, this is of course insufficient in order to conclude global existence and one needs to invoke the very delicate analysis of [47].
6.1
The nonradial cubic NLKG equation in R3
To precisely state the nonradial result from [111], let uE WD .u; u/ P and set ˚ H" WD uE 2 H j Em .E u/ < E.Q; 0/ C "2
(6.1)
with the minimal energy ˇ Em .E u/ WD ˇE.E u/2
ˇ1 u/2 P .E u/2 ˇ 2 sign E.E
P .E u/2 ;
(6.2)
where P .E u/ D hujrui P is the conserved momentum for the NLKG equation. In contrast to the energy E, the minimal energy Em is Lorentz invariant. It is therefore necessary to use this form of the energy in the general nonradial setting. We call a solution with zero momentum normalized. To every solution with jE.E u/j > jP .E u/j 2 there exists exactly one Lorentz transform which reduces it to a normalized one, see (2.83). If jE.E u/j < jP .E u/j, then there exists a Lorentz transform L such that E.E u ı L/ < 0. But such solutions are known to blowup in finite positive and negative times by the Payne–Sattinger criterion. If jE.E u/j D jP .E u/j, then along some sequence of Lorentz transforms Lj we have E.E u ı Lj / ! 0 as j ! 1. But then either K.u ı Lj / < 0 for some j , which means that u blows up in both time directions by the Payne–Sattinger criterion, or u ı Lj ! 0 strongly in H 1 so that u exists globally. Thus, we can always talk about normalized solutions in those cases where [114] does not apply, which is sufficient for our purposes. The 9-set theorem now takes the following form. Theorem 6.1. Consider all solutions of NLKG (2.1) with initial data uE .0/ 2 H" for some small " > 0. Then the solution set is decomposed into nine non-empty sets characterized as (1) Scattering to 0 for both t ! ˙1, (2) Finite time blow-up on both sides ˙t > 0, 2
At this point one needs to clarify the meaning of the Cauchy problem for Lorentz transformed solutions, and more generally, wellposedness questions of transformed solutions; this is perhaps not immediately clear as Strichartz norms such as L3t L6x are not invariant under Lorentz transforms. However, the norm L4t;x is, and is shown to control the relevant Strichartz norms in [111].
6.1 The nonradial cubic NLKG equation in R3
195
(3) Scattering to 0 as t ! 1 and finite time blow-up in t < 0, (4) Finite time blow-up in t > 0 and scattering to 0 as t ! 1, (5) Trapped by ˙Q for t ! 1 and scattering to 0 as t ! 1, (6) Scattering to 0 as t ! 1 and trapped by ˙Q as t ! 1, (7) Trapped by ˙Q for t ! 1 and finite time blow-up in t < 0, (8) Finite time blow-up in t > 0 and trapped by ˙Q as t ! 1, (9) Trapped by ˙Q as t ! ˙1, where “trapped by ˙Q” means that the normalized solution stays in a O."/ neighborhood of E C y/ j y 2 R3 g f˙Q. forever after some time (or before some time). The initial data sets for (1)–(4), respectively, are open. The approach is similar to the radial case, but with the added feature of having to control a small translation vector, after the Lorentz transform to the normalized solution. As for radial data, Theorem 6.1 extends to all Rd and equations of the form u C u D jujp
1
u;
1C
4 4
0 such that any even data .u0 ; u1 / 2 H 1 L2 .R/ with energy E.u; u/ P < E.Q; 0/ C "2
(6.35)
have the property that the solutions u.t/ of (6.32) associated with these data exhibit the following trichotomy: ı u blows up in finite positive time ı u exists globally and scatters to zero as t ! 1 ı u exists globally and scatters to Q, i.e., there exists a free Klein–Gordon wave v.t/; v.t/ P 2 H 1 L2 with the property that u.t/; u.t/ P D .Q; 0/ C v.t/; v.t/ P C oH .1/ t ! 1: (6.36) In addition, the set of even data as in (6.35) splits into nine nonempty disjoint sets corresponding to all possible combinations of this trichotomy as t ! ˙1. Moreover, we obtain a characterization of the threshold solutions, i.e., those with energies E.E u/ D E.Q; 0/, cf. [48]. In fact, we find the following. Corollary 6.4. The even solutions to (6.32) with energy E.E u/ D E.Q; 0/ can be characterized as follows: ı they blow up in both the positive and negative time directions ı they exist globally on R and scatter as t ! ˙1 ı they are constant ˙Q ı they equal one of the following solutions, for some t0 2 R: WC .t C t0 ; x/;
W .t C t0 ; x/; WC . t C t0 ; x/; W . t C t0 ; x/ where W˙ .t; /; @ t W˙ .t; / approach .Q; 0/ exponentially fast in H as t ! 1, and D ˙1. In backward time, WC scatters to zero, whereas W blows up in finite time. E form a one-dimensional stable manifold As usual, the images of WE˙ and Q associated with .Q; 0/, cf. [6]. The unstable manifold is obtained by time-reversal. Moreover, the solutions in Theorem 6.3 which scatter to Q form a C 1 manifold3 in 3
It is in fact smoother than C 1 .
206
6 Further developments of the theory
H of codimension 1 which is the center-stable manifold associated with .Q; 0/. The center manifold is obtained by the transverse intersection of the two center-stable manifolds corresponding to t ! ˙1, respectively. It therefore has codimension 2. We remark that the Hamza, Zaag [70] studied the problem of classifying blowup for equation (6.32) and showed that blowup occurs only in the form of ODE blowup for the one-dimensional NLKG equation (in higher dimensions the power on the nonlinearity needs to be no larger than then conformal power). The restriction p > 5 in Theorem 6.3 is most likely a technical one. We remark that it plays an important role in the adaptation of the “zero-frequency scattering result”, see Lemma 4.13. This was needed in the proof of the one-pass theorem to handle the degenerate case where the lower bound on K2 u.t/ becomes ineffective. In fact, it seems reasonable to expect that Theorem 6.3 remains valid in the range p > 3, and possibly even for some range p 3. At p D 3 the linearized operator LC has a threshold resonance which needs to be taken into account. But more importantly, in the range p < 5 any small data scattering argument cannot be based on Strichartz estimates alone which is of course a serious obstacle at this point. The power p D 5 is perhaps accessible, but we exclude it here (as in [77]). We remark that in contrast to the NLS equation, the hyperbolic structure underlying our proof of Theorem 6.3 is still present for all 1 < p 5. This comment refers to the fact that the structure of the spectrum of the linearized form of (6.32) around Q does not change significantly as one decreases p (whereas for NLS the exponentially stable eigenvalues merge with the root space at the L2 -critical power and then move off into the real discrete spectrum as one lowers the power further). See Bizo´n, Chmaj, Szpak [17] for a numerical study of (6.32) which exhibits different rates of convergence depending on the power of the nonlinearity. The proof of the one-dimensional “orbital stability”-type 9-set theorem in which one only requires trapping by .˙Q; 0/ rather than scattering to .˙Q; 0/, is very similar to the three-dimensional setting. The action (or static energy) is Z h i 1 1 .j@x uj2 C juj2 / jujpC1 dx J.u/ D (6.37) 2 pC1 and the functionals K0 ; K2 are now Z 0 K0 .u/ D hJ .u/jui D 0
K2 .u/ D hJ .u/jAui D
j@x uj2 C juj2 Z
j@x uj2
jujpC1 dx;
p 1 jujpC1 dx 2.p C 1/
with A D 21 .x@x C @x x/ being the generator of dilations. The linearizations of these
6.2 The one-dimensional NLKG equation
207
functionals are K0 .Q C v/ D
.p Dp
K2 .Q C v/ D
2 1/hQp jvi C O kvkH 1 ; E 5 p 2 Q C 2Qjv C O kvkH 1 :
2 Note the positivity in the linearization of K2 for p 5. The expansion of J is essentially the same as before with the linearized operator LC D @xx C1 pQp 1 : 1 3 J.Q C v/ D J.Q/ C hLC vjvi C O kvkH v ! 0: 1 ; 2 The translational symmetry in infinitesimal form becomes LC Q0 D 0 with 0 being a simple eigenvalue. Since Q0 has a single zero, it follows from Sturm oscillation theory that it is the second eigenfunction of LC , and one has LC D k 2 for some exponentially decaying > 0 as in the three-dimensional case. Over the even functions LC has no kernel. Moreover, the variational arguments which we developed in the previous chapters for the three-dimensional radial equation remain largely in effect. To be more specific, the respective statements remain intact but the arguments need to be adjusted; this is due to the absence of Strauss’ estimate in one dimension, cf. (2.20), and the subsequent loss of compactness which was used in the construction of various minimizers. However, this particular obstacle is easily circumvented by means of standard concentration-compactness arguments.
6.2.1 The center-stable manifold The center-stable manifold is again constructed by the Lyapunov–Perron method as in Chapter 3, and one obtains the following result. Proposition 6.5 (Center-Stable manifolds). Let p 5. There exists > 0 small and a C 1 graph M in B .Q; 0/ H so that .Q; 0/ 2 M, with tangent plane ˚ TQ M D .u0 ; u1 / 2 H j hku0 C u1 ji D 0 at .Q; 0/ in the sense that sup
dist.x; TQ M/ . ı 2 ;
80 0. Under our assumptions on V they are given by the Volterra
209
6.2 The one-dimensional NLKG equation
equation 1
fC .x; z/ D e
iz 2 x
1
Z
1
sin .x
y/z 2 1
z2
x
V .y/fC .z; y/ dy
and similarly for f .x; z/, for all Im.z/ 0. Finally, the Green function is given by the expression, for x > x 0 and Im z > 0, .H
z/
1
.x; x 0 / D
fC .x; z/f .x 0 ; z/ ; W .z/
W .z/ D W fC .; z/; f .; z/
where the latter is the Wronskian (the case x 0 > x being analogous by symmetry). It is not hard to see that W .z/ ¤ 0 for all Im z 0, z ¤ 0, whence .H z/ 1 remains bounded as in (3.75) with > 21 away from z D 0. The only possible failure of boundedness therefore occurs at z D 0. Moreover, the resolvent blows up as z ! 0 if and only if W .0/ D 0 which is equivalent to the solutions f˙ .; 0/ being linearly dependent; in other words, if and only if zero energy is a resonance (it also follows immediately that zero energy cannot be an eigenvalue for V as above). We now summarize some basic spectral properties of LC and L WD @xx C 1 Qp 1 . Lemma 6.6. If p > 3, then LC and L have the following properties: they have no eigenvalues in the interval .0; 1 and for both LC and L the threshold 1 is not a resonance. Furthermore, LC has exactly two eigenvalues, viz. k 2 and 0, and L has exactly one, namely 0. Proof. This is a classical observation based on the fact that for sech2 .x/ potentials (which go by the name of Pöschl–Teller potentials) the eigenvalue equations can be explicitly integrated via hypergeometric functions. For example, the lemma can easily be deduced from Flügge [55], Problem 39, page 94. See [90], Lemma 9.1 for more details.
6.2.2 The distorted Fourier transform We will obtain the desired local estimate (6.38) on the KG-evolution of L˙ by means of the distorted Fourier transform. We begin by recalling the distorted Fourier transform relative to a general Schrödinger operator on the line L WD
d2 CV dx 2
(6.39)
210
6 Further developments of the theory
with real-valued potential V . In our specific application, V .x/ D
˛ cosh
2
(6.40)
.ˇx/
for suitable ˛; ˇ > 0, but for the moment we only assume that V 2 L1loc .R/ and that L is limit-point at ˙1. This material is of course standard, see for example Section 2 of [59]. It is simpler to start with the operator L from (6.39) on the half-line x 0. This requires that we impose a boundary condition at x D 0, say (for simplicity) the Dirichlet boundary condition. In what follows, we avoid discussion of domains and self-adjointness of various operators, see [59] for more on these issues. Let .x; z/; .x; z/ be a fundamental system with .0; z/ D 0 .0; z/ D 0; 0 .0; z/ D .0; z/ D 1 : Note that W .; z/; .; z/ D 1. By the limit-point assumption at x D 1 there 2 exists a nonzero solution .; z/ 2 L .0; 1/ for every z 2 C n R, unique up to nonzero complex multiples. The basic relation of Weyl theory is the identity d W . .; z1 /; dx
; z2 / .x/ D .z2
z1 / .x; z1 / .x; z2 /;
z1 ; z2 2 C n R :
Integrating it from 0 to 1 yields, with z2 D z 1 , z1 D z, 1
Z 0
ˇ ˇ W ˇ .x; z/ˇ2 dx D
.; z/; .; z/ .0/ : 2i Im z
Since .; z/ D c .; z/ with some c ¤ 0, we conclude that particular, we can normalize .; z/ in such a way that
(6.41) .0; z/ ¤ 0. In
.; z/ D .; z/ C m.z/.; z/ : Indeed, .0; z/ D 1 and m.z/ D 0 .0; z/. By the preceding, m is analytic on Im z > 0 and satisfies Im m.z/ > 0 for those arguments. In fact, from (6.41), Z 1 ˇ ˇ ˇ .x; z/ˇ2 dx D Im m.z/ : Im z 0 In other words, m is a Herglotz function and thus admits the representation Z h i 1 .d/ m.z/ D Re m.i/ C z 1 C 2 R
211
6.2 The one-dimensional NLKG equation
for some positive measure . The latter is called the spectral measure of H and it can be found as follows: Z 2 Cı (6.42) Im m. C i"/ d: Œ1 ; 2 / D 1 lim lim ı!0C "!0C 1 Cı
The measure appears naturally in the unitarity of the distorted Fourier transform which we now define. For any Schwartz function f set Z 1 fO./ D f .x/.x; / dx; 8 0 : 0
Then for any Schwartz f; g one has the “Plancherel identity” ˝ ˛ ˝ ˛ f j F .H /.1 ;2 .H / g L2 ..0;1// D fO j F .1 ;2 b g L2 .R;/
(6.43)
where F is continuous, bounded, and 0 < 1 < 2 < 1 are arbitrary. The proof of (6.43) is simple, and follows most easily from Stone’s formula which relates the spectral resolution E./ of a self-adjoint operator to the resolvent, i.e., ˝
Z 2 Cı ˛ 1 f; E Œ1 ; 2 / g D lim lim 2 i ı!0C "!0C 1 Cı ˝ f; .H . C i"//
1
.H
.
i"//
1
˛ g d ;
see [59]. For the resolvent (Green function) one has the explicit expression in terms of ; , viz. if 0 < x < x 0 and Im z > 0, say, then .H
z/
1
.x; x 0 / D .x; z/ .x 0 ; z/
and symmetrically if 0 < x 0 < x. By means of this formula, Stone’s representation, and (6.42), one obtains (6.43) by explicit calculation. For the free half-line problem (i.e., if V D 0) the reader will easily check that 1 1 m.z/ D iz 2 , ./ D 1 2 d. In that case (6.43) is nothing but the standard Plancherel theorem for the Sine-transform on the half-line. In general, note that (6.43) extends by unitarity to all of L2 .0; 1/ , respectively L2 .R; /. In particular, it encompasses the entire spectrum including eigenvalues. We now turn to developing these ideas for the full line, which is again standard. The main difference is that one is now dealing with multiplicity two, and a spectral measure which therefore takes its values in the nonnegative 2 2-matrices. For the full-line we need to assume that both ends ˙1 are limit-point. Define ˛ .x; x0 I z/, ˛ .x; x0 I z/ to be the fundamental system of solutions of L
Dz ;
z2C
212
6 Further developments of the theory
so that ˛0 .x0 ; x0 I z/ D
˛ .x0 ; x0 I z/ D ˛0 .x0 ; x0 I z/
sin ˛;
(6.44)
D ˛ .x0 ; x0 I z/ D cos ˛
where x0 2 R and ˛ 2 Œ0; /. Their Wronskian is W .˛ ; ˛ / D 1 : The Weyl–Titchmarsh solutions are defined as the unique solutions ˙;˛ .; x0 I z/ 2 L2 .Œx0 ; ˙1/; dx/ for z 2 C n R which satisfy the boundary condition 0 ˙ .x0 ; x0 I z/ sin ˛
C
˙ .x0 ; x0 I z/ cos ˛
D 1:
This boundary condition ensures that ˙;˛ .x; x0 I z/
(6.45)
D ˛ .x; x0 I z/ C m˙;˛ .z; x0 /˛ .x; x0 I z/
and the Wronskian W.
C .; x0 I z/;
.; x0 I z// D m
;˛ .z; x0 /
mC;˛ .z; x0 / :
The Weyl–Titchmarsh functions m˙;˛ are Herglotz functions, and the associated Weyl–Titchmarsh matrix 3 2 1 m ;˛ .z;x0 /CmC;˛ .z;x0 / 1 M˛ .z; x0 / WD 4
m 1m 2m
;˛ .z;x0 /
mC;˛ .z;x0 /
2m
;˛ .z;x0 /CmC;˛ .z;x0 / ;˛ .z;x0 /
mC;˛ .z;x0 /
m m
;˛ .z;x0 /
mC;˛ .z;x0 /
;˛ .z;x0 /mC;˛ .z;x0 / ;˛ .z;x0 /
5
(6.46)
mC;˛ .z;x0 /
is a Herglotz matrix. This implies that there exists a nonnegative 2 2-matrix-valued measure ˝˛ .d; x0 / so that the representation Z h 1 i M˛ .z; x0 / D C˛ .x0 / C ˝˛ .d; x0 / z 1 C 2 R
(6.47) Z
˝˛ .d; x0 / C˛ .x0 / D C˛ .x0 / ; 0. The O./-terms satisfy the natural derivative bounds. Proof. Taking ˛ D 0 and x0 D 0 in (6.44) (and suppressing x0 ) yields 0 .xI z/ D 0 . xI z/;
0 .xI z/ D
0 . xI z/;
;0 .xI z/
D
C;0 .
xI z/
whence m ;0 .z/ D mC;0 .z/ and W .z/ D 2m ;0 .z/. Denote by u0;C .x/ and u1;C .x/ a fundamental system of solutions to Lf D 0 with u0;C .x/ D 1 C O.x
100
u1;C .x/ D x C O.x
100
/
(6.53)
/
as x ! 1. This representation follows from the Volterra integral equations Z 1 u0;C .x/ D 1 C .y x/V .y/u0;C .y/ dy ; Zx 1 u1;C .x/ D x C .y x/V .y/u1;C .y/ dy x
by iteration. In particular, W .u1;C ; u0;C / D 1. Furthermore, u0; ; u1; denote the corresponding solutions, but with x ! 1. By symmetry, u0; .x/ D u0;C . x/ and u1; .x/ D u1;C . x/. Since zero energy is nonresonant, W .u0;C ; u0; / ¤ 0. Perturbatively in , we now obtain from uj;C .x/ unique eigenfunctions uj;C .x; / satisfying Luj;C D uj;C , as well as for small and jx 2 j 1, uj;C .x; / D uj;C .x/ 1 C O.x 2 / ; j D 0; 1 : Indeed, uj;C .x; / are given in terms of the Volterra equations Z x uj;C .x; / D uj;C .x/ C u0;C .x/u1;C .y/ u0;C .y/u1;C .x/ uj;C .y; / dy : 0
Similarly, the Jost solutions fC .x; / defined by Lf˙ .; / D f˙ .; /, f˙ .x; / 1
' e ˙ix 2 as x ! ˙1, satisfy 1
f˙ .x; / D e ˙ix 2 1 C O.x
100
/ ;
˙x 1 :
215
6.2 The one-dimensional NLKG equation
As usual, this follows from the Volterra representation of these functions. Moreover, one has f˙ .x; / D a˙ ./u0;˙ .x; / C b˙ ./u1;˙ .x; / with W f˙ .; /; u1;˙ .; / ;
a˙ ./ D
b˙ ./ D W f˙ .; /; u0;˙ .; / :
Therefore, for any small " > 0 a˙ ./ D 1 C O.1 " / ;
(6.54)
1
b˙ ./ D i 2 C O.1 " / as ! 0. In conclusion, 1 W ./ WD W fC .; /; f .; / D c0 C ic1 2 C O.1 " /
(6.55)
where c0 ; c1 2 R with c0 ¤ 0; this latter nonvanishing is precisely the nonresonant condition at zero energy. The matrix in (6.46) is M0 ./ D diag W ./
1
;
1 W ./ 4
and the measure ˝0 ./ satisfies, for small , by (6.48) 1 1 ˝0 .d/ D diag O. 2 /; O. 2 / d;
! 0:
(6.56)
For large , the free representation (6.51) describes ˝0 to leading order. The regularity of the spectral measure from Lemma 6.8 implies the following improved estimates on the time evolution. These bounds are crucial in the nonlinear analysis near the ground states. Note that (6.57) cannot hold for the free case, i.e., 1 1 2 e i th@x i 2 since the best pointwise decay of the latter evolution is t 2 as can be seen from the pointwise analysis in the final section of Chapter 2. Lemma 6.9. Let L˙ be as above, with p > 3. Then one has the following bounds
hxi
hxi
1
Z
e 1
t
e 1
1
2 1 ˙itL˙
2 ˙i.t s/L˙
Pc .L˙ /f
2 L1 x Lt
Pc .L˙ /f .s/ ds
for all Schwartz functions f on the line.
2 L1 x Lt
1 . h@x i 2 f 2 Z
1
. 1
h@x i 21 f .s/ ds 2
(6.57)
216
6 Further developments of the theory
Proof. We begin with the first. Write w.x/ D hxi 1 . Then by duality we need to estimate, with g D g.t; x/, and F denoting the distorted Fourier transform of Proposition 6.7, Z 1 1 E Z 1 1 ˇ 2 e ˙i tL˙ Pc .LC /f ˇ wg D dt e ˙it.1C/ 2 F f ./T ˝0 .d/F wg.t/ ./ 1 0 Z 1 Z 1 Z 1 1 ˙it.1C/ 2 F1 f ./1 ./ D dt e w.x/g.t; x/.x; / dx d 1 0 1 Z 1 Z 1 Z 1 1 C dt e ˙it.1C/ 2 F2 f ./2 ./ w.x/g.t; x/.x; / dx d
D
0
1
1
(6.58)
where ˝0 .d/ D diag.1 ; 2 / d, and ; are 0 .x; 0I z/, 0 .x; 0I z/ from above. Here denotes the spectral variable of L D L˙ 1. For small one has 1 1 .1 C / 2 D 1 C C O.2 / : 2 Up to change of variable in (which we ignore) we can therefore view the small integral as the usual Fourier transform. Denoting the Fourier transform of g.t; x/ in time by b g .; x/ we can bound the contribution of small to (6.58) by means of (6.56) as follows: 1
Z 0
Z
1 1
ˇ ˇˇ ˇ ˇFj f ./ˇ ˇb g .; x/ˇ j.x; /j C j.x; /j w.x/dx j ./ d
1 . sup sup w.x/j ./ 2 j.x; /j C j.x; /j 0 1, one has hi 2 ' 2 . Hence, the first component for these -values is
6.2 The one-dimensional NLKG equation
bounded by (we can ignore w.x/ in this regime, as well as , cf. (6.51)), Z 1Z 1 ˇ ˇˇ p ˇ ˇF1 f ./ˇˇb g . ; x/ˇ1 ./ dxd 1 Z 11 Z 1 ˇ ˇˇ ˇ ˇF1 f .2 /ˇˇb . g .; x/ˇ 1 .2 / dxd 1 1 Z 1 Z 1 21 Z 1 ˇ ˇ ˇ2 2 ˇ2 21 ˇ ˇ ˇ . F1 f ./ 1 ./ d b g .; x/ˇ d dx 1
1
217
(6.60)
1
1 . h@x i 2 f 2 g L1 L2 : x
t
The final estimate here follows by the free asymptotics (6.51). For the second inequality in the lemma one proceeds in a similar fashion. In fact, using the notation from the first part of the proof, 1 E ˇ 2 e ˙i.t s/L˙ Pc .L˙ /f .s/ ds ˇ wg 1 Z 1 Z t Z 1 1 D dt ds e ˙i.t s/.1C/ 2 F f .s/ ./T ˝0 .d/F wg.t/ ./ 1 1 1 Z 1 Z 1 Z 1 1 D ds dt e ˙i.t s/.1C/ 2 F1 f .s/ ./1 ./ 1 s 0 Z 1 w.x/g.t; x/.x; / dx d 1 Z 1 Z 1 Z 1 1 C ds dt e ˙i.t s/.1C/ 2 F2 f .s/ ./2 ./ 1 s 0 Z 1 w.x/g.t; x/.x; / dx d :
DZ
t
1
Carrying out the t-integration, and performing similar arguments as in the previous case, shows that the two final expressions here are Z 1
h@x i 12 f .s/ ds g 1 2 . 2 L L 1
x
t
as desired. By means of a suitable interpolation argument, one now obtains the following result, see [88].
218
6 Further developments of the theory
Corollary 6.10. Any solution of LC u C F; u ? ; Q0
uR D
(6.61)
satisfies
hxi s u 2 1=2 . uŒ0 1 2 C F 1 2 L L H L L H t
x
t
x
(6.62)
with s > 32 . The significance of (6.62) lies with the perturbative nonlinear analysis near the ground states. In fact, placing a nonlinear term such as Qq v 2 in L1t L2x yields
q 2
Q v 1 2 . hxi s v 2 2 4 . hxi s v 2 1 (6.63) L t Lx L t Lx 2 2 L t Hx
for H 1 solutions v. This observation allows one to close all estimates very easily. In addition, one of course also uses Strichartz estimates for the Klein–Gordon equation relative to L (in order to estimate v p , say). Note that unlike Lemma 6.9, the following result has nothing to do with L having a zero energy resonance or not. Corollary 6.11. Let L be as in (6.39). For any Schwartz function u in R1C1 t;x with u D Pc .L/u one has the aprori bound
u p r . uŒ0 1 2 C uR C Lu 1 2 (6.64) L L H L L L x
t
for any 4 < p 1, 0 < 5 p < 1.
1 r
t
1 2
2 . p
x
In particular, one can take r D 2p for any
The proof is a simple application of the Lp -boundedness of the wave operators, see [3], [141] (note that this boundedness property does not require zero energy to be regular) and the Strichartz estimates for the free KG equation, see Exercise 2.45. For all remaining details, we refer the reader to [88].
6.3
The cubic radial NLS equation in R3
In this section as well as the next we turn to other equations than NLKG. More specifically, following [110], we develop similar results for the cubic NLS equation in R3 with radial data, viz. i@ t u
u D juj2 u;
.t; x/ 2 R1C3 :
(6.65)
6.3 The cubic radial NLS equation in R3
219
The local well-posedness of (6.65) in the energy space H 1 is classical, see for example Strauss [133], Sulem, Sulem [135], Cazenave [27], and Tao [136]. One has mass and energy conservation
1
u 2 D const:; 2 2
2 1 4 1
u D const:; E.u/ D ru 2 4 2 4
M.u/ D
(6.66)
where k kp denotes the Lp .R3 / norm. Data with small H 1 norm have globally defined solutions which scatter to a free wave. For the defocusing equation it is known that all energy solutions scatter to zero, see Ginibre, Velo [61]. In contrast, (6.65) is known to exhibit energy data for which the solutions blow up in finite time. In fact, Glassey [62] proved that all data of negative energy are of this type provided they also have finite variance kxuk2 < 1. The latter assumption was later removed for radial solutions by Ogawa, Tsutsumi [113]. Eq. (6.65) possesses a family of special oscillatory solutions of the form 2 u.t; x/ D e i t ˛ Ci Q.x; ˛/ where ˛ > 0 and Q.; ˛/ C ˛ 2 Q.; ˛/ D jQj2 Q.; ˛/ : As in Chapter 2, the ground state is singled out as the unique radial positive solution to this equation. Letting modulation and Galilean symmetries act on these special solutions u.t; x/ generates an eight-dimensional manifold of solitons. In the radial context, the manifold is only two-dimensional. The question of orbital stability of these solitons in the energy space was settled by Weinstein [143], [144], Berestycki, Cazenave [11], and Cazenave, Lions [28]. A general theory which covers this example – as well as many others such as the NLKG equation – was developed by Grillakis, Shatah, Strauss [67]. The cut-off in the power jujp 1 u in the n-dimensional setting turns out to be the L2 critical one p0 D n4 C 1, with p p0 being unstable and p < p0 stable. In particular, the cubic NLS (6.65) is unstable. Holmer, Roudenko [74] showed that for all radial solutions u with mass kuk2 D kQk2 and energy E.u/ < E.Q/ there is the following dichotomy: if kruk2 < krQk2 one has global existence and scattering (as jtj ! 1), whereas for kruk2 > krQk2 there is finite time blowup in both time directions. The radial assumption was then removed in Duyckaerts, Holmer and Roudenko [46]. Note that the mass condition is easily removed by scaling, with M.u/E.u/ being the natural scaling-invariant version of the energy, and with M.u/kruk22 replacing kruk22 . For the subcritical NLKG equation the analogous results were obtained in [77]. In analogy with NLKG, it follows from the variational properties of Q that these regions
220
6 Further developments of the theory
are invariant under the NLS flow. The methods in the three aforementioned papers follow the ideology of Kenig–Merle [84] in order to establish scattering. The second author began the investigation of the conditional4 asymptotic stability problem for focusing dispersive equations in [122] by means of the equation (6.65) but with general, rather than radial, data. While a strong, and noninvariant topology was used there, Beceanu [9] later established the corresponding 1 result in the optimal topology HP 2 .R3 /; in Chapter 3 we prove a special case of Beceanu’s theorem, see Proposition 3.31. In one dimension, Krieger and the second author [90] considered the one-dimensional NLS equation, and established the desired conditional asymptotic stability in that case, but again in a more restrictive topology than the energy space. Finally, we note that the work of Bates and Jones [6] was shown to apply to the NLS equation as well, see Gesztesy, Jones, Latushkin, Stanislavova [58]. A major difference from the NLKG equation is that here one has the “cone” of solitons (under the radial restriction) [ S˛ WD fe i Q.; ˛/ j 2 Rg; S WD S˛ : ˛>0
We set Q D Q.; 1/ for convenience. Then Q.x; ˛/ D ˛Q.˛x/ and M Q.; ˛/ D ˛ 1 M.Q/. The first result from [110] is again the 9-set one. To be specific, let 1 H D Hrad .R3 / and ˚ (6.67) H" WD u 2 H j M.u/E.u/ < M.Q/ E.Q/ C "2 as well as ˚ H"˛ WD H" \ u 2 H j M.u/ D M Q.; ˛/
(6.68)
for any ˛ > 0. Theorem 6.12. There exists " > 0 small such that all solutions of (6.65) with data in H"1 exhibit one of the following nine different scenarios, with each alternative being attained by infinitely many data in H"1 : (1) Scattering to 0 for both t ! ˙1, (2) Finite time blowup on both sides ˙t > 0, (3) Scattering to 0 as t ! 1 and finite time blowup in t < 0, (4) Finite time blowup in t > 0 and scattering to 0 as t ! 1, 4
This refers to the fact that the data are restricted to lie on a manifold of finite codimension.
6.3 The cubic radial NLS equation in R3
221
(5) Trapped by S1 for t ! 1 and scattering to 0 as t ! 1, (6) Scattering to 0 as t ! 1 and trapped by S1 as t ! 1, (7) Trapped by S1 for t ! 1 and finite time blowup in t < 0, (8) Finite time blowup in t > 0 and trapped by S1 as t ! 1, (9) Trapped by S1 as t ! ˙1, where “trapped by S1 ” means that the solution stays in a O."/ neighborhood of S1 relative to H 1 forever after some time (or before some time). The initial data sets for (1)–(4), respectively, are open in H"1 . The set of data in H 1 for which the associated solutions of (6.65) forward scatter, i.e., .1/ [ .3/ [ .6/, is open, pathwise connected, and unbounded; in fact, it contains curves which connect 0 to 1 in H 1 . The theorem applies to solutions of any mass by rescaling. More precisely, if u 2 H"˛ , then the statement remains intact with S1 replaced by S˛ and “trapped” by S˛ now meaning that dist.u; S˛ / . " where the distance is measured in the metric k kH˛1 WD ˛
1
k k2HP 1 C ˛k k22
21
:
(6.69)
As for the NLKG equation, the main ingredient for Theorem 6.12 is a suitable “onepass theorem”. It precludes almost homoclinic orbits which start very close to S1 and eventually return very close to S1 . In combination with an analysis of the hyperbolic dynamics near S1 which results from the exponentially unstable nature of the ground state solution, this allows one to show that the fate of the solution is governed by a virial-type functional K after it exits a neighborhood of S1 . Invoking some finer spectral properties of the Hamiltonian obtained by linearizing the NLS equation around Q, we obtain the following stronger statement which describes in more detail what “trapping” means. As for the NLKG equation, the relevant notion is that of a center-stable manifold. In fact, we constructed such a manifold in Chapter 3 for the NLS equation in the radial energy topology, see Proposition 3.31. For the following theorem it is advantageous not to freeze the mass. In other words, we work with the full set H" . We require the following terminology: Definition 6.13. Let u.0/ 2 H" define a solution u.t/ of (6.65) for all t 0. We say that u forward scatters to S iff there exist continuous curves W Œ0; 1/ ! R and ˛ W Œ0; 1/ ! .0; 1/, as well as u1 2 H such that for all t 0 u.t/ D e i.t/ Q ; ˛.t/ C e
it
u1 C ˝.t/
where k˝.t/kH 1 ! 0 as t ! 1, ˛.t/ ! ˛1 > 0 as t ! 1.
(6.70)
222
6 Further developments of the theory
Note that one then necessarily has M.u/ D M Q.; ˛1 / C M.u1 / D ˛11 M.Q/ C M.u1 /;
2
2 1 1 E.u/ D E Q.; ˛1 / C ru1 2 D ˛1 E.Q/ C ru1 2 2 2 whence (using that E.Q/ D M.Q/ > 0), ku1 k22 kru1 k22 2"2 ; 2M.Q/ E.u/ ; E.Q/
˛11 kru1 k22 C ˛1 ku1 k22 C M.Q/ ˛1 M.u/
(6.71)
(6.72)
and in particular, we conclude that ku1 kH˛1 ", that ˛1 is bounded from both 1 above and below, and that M.u/E.u/ M.Q/E.Q/. The heuristic meaning of (6.70) is simply that u asymptotically decomposes into a soliton e i1 .t/ Q.; ˛1 / plus an H 1 -solution to the free Schrödinger equation (however, the phase 1 is not precisely the one associated with Q.; ˛1 / which 2 C 1 ). In fact, in those cases where one can establish (6.70) it would mean t˛1 is possible to obtain finer statements on and ˛, see [110]. Theorem 6.14. Using the conclusion of Proposition 3.28 one has the following. There exists " > 0 small such that all solutions of (6.65) with data in H" exhibit one of the nine different scenarios described in Theorem 6.12, provided we replace “trapped by S1 ” with “scattering to S". Moreover, each alternative is attained by infinitely many data in H" . The sets .5/ [ .7/ [ .9/ and .6/ [ .8/ [ .9/ are smooth codimension-1 manifolds in the phase space H. Similarly, .9/ is a smooth manifold of codimension two, and it contains S. Using5 the terminology of Chapter 3 we can say that .5/ [ .7/ [ .9/ and .6/[.8/[.9/ are the center-stable manifold Mcs , resp. the center-unstable manifold Mcu , associated with Q – modulo the symmetries given by ˛ and . Since center manifolds are in general not unique it might be more precise to say “a center-stable manifold” here. However, our manifolds are naturally unique for the global characterization in Theorem 6.12. Similarly, .9/ is the center manifold of Q, again modulo the symmetries given by ˛ and . 5
This is somewhat of an abuse of language, since the theory presented there did not include symmetry parameters. However, the Lyapunov–Perron approach as it appears towards the end of that chapter also applies when symmetries are present, see [122], [9] etc. Furthermore, the graph transform is also know to apply in the presence of symmetries, at least for the NLKG equation, see [112].
223
6.3 The cubic radial NLS equation in R3
Every point p 2 S has a neighborhood B" .p/ of size . " relative to the metric (6.69) with ˛ D M.Q/=M.p/, such that B" .p/ is divided by Mcs into two connected components; all data in one component lead to finite time blow-up for positive times, whereas all data in the other lead to global solutions for positive times which scatter to zero as t ! C1. All solutions starting on Mcs itself scatter to S in the sense of (6.70) as t ! C1. As is to be expected by analogy with the NLKG equation and the work of Duyckaerts and Merle [48], the one-dimensional stable and unstable manifolds (up to the symmetries) appear naturally in the form of those solutions found by Duyckaerts and Roudenko [49]. It is important to note that we can therefore completely describe the global (i.e., both as t ! 1 as well as t ! 1) behavior of the stable/unstable manifolds in this setting. Theorem 6.15. Consider the limit " ! 0 in Theorem 6.12, i.e., all the radial solutions satisfying E.u/ E.Q/ and M.u/ D M.Q/. Then the sets (3) and (4) vanish, while the sets (5)–(9) are characterized, with some special solutions W˙ of (6.65), as follows: .5/ D fe i W .t i
.7/ D fe WC .t .9/ D fe
i.t C /
t0 / j t0 ; 2 Rg; t0 / j t0 ; 2 Rg;
.6/ D fe i W . t i
.8/ D fe WC . t
t0 / j t0 ; 2 Rg; t0 / j t0 ; 2 Rg; (6.73)
Q j 2 Rg:
The sets .5/ [ .7/ [ .9/ form the stable manifold, whereas .6/ [ .8/ [ .9/ are the unstable manifold of Q, up to the modulation symmetry. In other words, solutions in .5/; .7/ and .6/; .8/ approach a soliton trajectory in S1 exponentially fast as t ! 1 or t ! 1, respectively. An analogous ˚ statement holds without the mass constraint, but then these sets take the form e i ˛W˙ .˛ 2 .t t0 /; ˛x/ , ˚ i ˚ 2 e ˛W˙ . ˛ 2 .t C t0 /; ˛x/ , resp. e i.t˛ C/ Q.; ˛/ , where ; ˛ vary. As in the case of Theorem 6.14 this again relies on both the spectral description of Proposition 3.28 as well as the center-stable manifold constructed in Theorem 3.31.
6.3.1 Some elements of the proofs for the NLS equation While there are many similarities with the NLKG equation studied in the previous chapters, there are also some essential differences. As is apparent from the statements of the main results, see Theorems 6.12–6.15 above, one major difference lies
224
6 Further developments of the theory
with the presence of the symmetry parameters given by scaling and the modulation of the phase. While the former can be frozen (at least for the orbital stability arguments leading to Theorem 6.12), the latter is removed simply by “modding out” the phase. More precisely, to analyze the dynamics near the ground states one sets u D e i .Q C w/, and then decomposes w further into the discrete modes of the linearized Hamiltonian, plus the dispersive part. Note that the latter also contains the root modes, i.e., those (generalized) eigenfunctions of the linearized Hamiltonian with zero energy. As to be expected, the root modes are essentially removed by means of suitable orthogonality conditions. Variational and linearized structures. To be more specific, we use the functionals E.u/ D kruk22 =2
kuk44 =4;
J.u/ D kruk22 =2 C kuk22 =2 3 K.u/ D kruk22 kuk44 ; 4
M.u/ D kuk22 =2; kuk44 =4;
the first three being the conserved energy, mass, and action, respectively. The functional K results from pairing J 0 .u/ with .xr C rx/u=2, the generator of dilations. By construction, Q is a critical point of J , i.e., J 0 .Q/ D 0 whence also K.Q/ D 0. Moreover, the region M.u/E.u/ < M.Q/E.Q/
(6.74)
is divided into two connected components by the conditions fK 0g and fK < 0g in analogy with the Payne, Sattinger sets P S˙ for NLKG. The quantity ME in (6.74) is scaling invariant and was used by Holmer, Roudenko [74] in their scattering analysis. The aforementioned division into two connected components is intimately linked to the following minimization property. Define positive functional G and I by G.'/ WD J.'/ I.'/ WD J.'/
K.'/ 1 1 2
' 2 2 ; D r'kL 2 C L 3 6 2
K.'/ 1 1 4 2 D 'kL ' L4 : 2 C 2 2 8
As already noted in Chapter 2 one has a variational characterization of the ground state. The only difference here is the functional I.'/ which is needed in the proof of the one-pass theorem.
225
6.3 The cubic radial NLS equation in R3
Lemma 6.16. We have J.Q/ D inffJ.'/ j 0 6D ' 2 H 1 ; K.'/ D 0g D inffG.'/ j 0 6D ' 2 H 1 ; K.'/ 0g D inffI.'/ j 0 6D ' 2 H 1 ; K.'/ 0g; and these infima are achieved only by e i Q.x
c/, with 2 R and c 2 R3 .
The NLKG equation does not admit the scaling symmetry due to the mass term, i.e., the u which is added to u. In contrast, for NLS one does need to consider the scaled family of ground states Q.˛/ D Q˛ WD ˛Q.˛x/ which solve Q˛ C ˛ 2 Q˛ D Q˛3 ; krQ˛ k22 D ˛krQk22 ;
kQ˛ k44 D ˛kQk44 ;
kQ˛ k22 D ˛
1
kQk22 :
Differentiating in ˛ yields . C ˛2
3Q˛2 /Q˛0 D
2˛Q˛ :
To commence with the perturbative analysis, one makes the ansatz u D e i .Q C w/: Inserting this into NLS yields P .u C juj2 u C u/ P D . C /.Q C w/ C jQj2 Q C 2jQj2 w C Q2 w C 2Qjwj2 C w 2 Q C jwj2 w
i wP D e
i
P D .1 C /.Q C w/
Lw C N.w/ ; (6.75)
where N.w/ is the nonlinear part defined by N.w/ D 2Qjwj2 C Qw 2 C jwj2 w and the linearized operator Lw WD
w C w
2Q2 w
Q2 w ;
is considered as an R-linear operator L. It is self-adjoint on L2 .R3 I C/ with the inner product Z hf jgi WD Re f .x/g.x/ dx : (6.76) R3
226
6 Further developments of the theory
The extension of L to a complex-linear operator was used in [110] only for the implementation of the Lyapunov–Perron method. This was already carried out at the end of Chapter 3, see Section 3.5. The point is that the dispersive theory required in the Lyapunov–Perron approach is most conveniently carried out in the matrix formalism corresponding to a complex-linear rendition of L, see Section 3.4.4. The natural symplectic form in this context is Z f .x/g.x/dx D hif jgi ˝.f; g/ WD Im R3
and i L is symmetric with respect to ˝, i.e., ˝.i Lf; g/ D ˝.iLg; f /. Since L is not self-adjoint as a complex operator,it posses generalized eigenfunctions which are not eigenfunctions (in other words, the Jordan form has a nilpotent part). Indeed, i LiQ D 0;
iLQ0 D
iLG˙ D ˙G˙ ;
2iQ;
where > 0, Q0 D @˛ Q˛ j˛D1 D .1 C r@r /Q; and with '; are
G˙ D ' i ;
real-valued. In terms of the real and imaginary values, these equations LC Q0 D
L Q D 0;
2Q;
D ';
L
LC ' D
with L D The existence of ';
C1
Q2 ;
LC D
C1
3Q2 :
is standard and follows from the minimization p ˚˝p ˛ L LC L f jf j kf k22 1 < 0; min
see for example [135] or [122]. Since Q > 0 is the ground state of L . Thus, 2 L 0 and ker.L / D fQg. In other words, hL f jf i & kf kH 1 if f ? Q. After appropriate normalization of .'; /, we have hi iQjQ0 i D
hQjQ0 i D M.Q/; 0
hi GC jG i D 2h'j i D 2hL
j i= D 1;
0
0 D hQjG˙ i D hiQ jG˙ i D h'jQi D h jQ i: Moreover h jQi 6D 0 and so we can choose h jQi > 0. To see this, suppose ? Q, then ' ? LC Q D 2Q3 , and so by Lemma 3.2, 0 hLC 'j'i D h j'i < 0, which is a contradiction.
227
6.3 The cubic radial NLS equation in R3
The symplectic decomposition of L2 .R3 I C/ corresponding to these discrete modes is uniquely given by f D aiQ C bQ0 C cC GC C c G C ; a D hif jQ0 i=M.Q/;
bD
hf jQi=M.Q/;
c˙ D ˙hif jG i :
One has 0 D hjQi D hijQ0 i D hijG˙ i and the symplectic projections onto fiQ; Q0 gi? and onto fG˙ gi? commute. We apply the symplectic decomposition to w. Then, writing D aiQ CbQ0 C one has u D e i .Q C w/ D e i .Q C C GC C G C / : The justification for including the “root”-part (i.e., the zero modes) in follows from a suitable choice of the symmetry parameters ˛; , as we shall see below. The action is expanded as J.u/
J.Q/ D
1 hLwjwi 2
C.w/ D
1 C C hL j i 2
C.w/ ;
(6.77)
where the superquadratic part C.w/ is defined by ˝ ˛ C.w/ D jwj2 wjQ C kwk44 =4 : The following lemma will guarantee the positivity of the component in (6.77). Lemma 6.17. Let f; g 6D 0 be real-valued, radial and satisfy ˝ ˛ ˝ ˛ f j D 0 D gjQ0 : 2 2 Then hLC f jf i ' kf kH 1 and hL gjgi ' kgkH 1 .
We leave the proof as an exercise, see [110]. Modulation theory. The modulation theory needed to implement the 9-set theorem for NLS is considerably easier than the one which was used in the proof of Theorem 3.31. This is due to the fact that while the latter addresses asymptotic stability issues, the former deals with orbital stability. To be precise, one determines ˛; by means of M.u/ D M.Q˛ /;
.uje i Q˛0 / < 0 ;
(6.78)
228
6 Further developments of the theory
where .f jg/ D
R
f .x/g.x/ dx. Note that an explicit solution is given by D Im log.uj
˛ D M.Q/=M.u/;
Q˛0 / :
Since .Q˛ jQ˛0 / D ˛ 2 M.Q/ < 0, there is a unique solution .˛; e i / 2 .0; 1/ S 1 as long as u is close to some e i Q˛ . It is easy to see that u D e i' Qˇ gives ' D and ˛ D ˇ. One advantage of this explicit choice of .˛; / is that M.u/ is conserved in time, and so ˛ is fixed. On the other hand, it is nonlinear in the sense hwjQ˛ i D
M.w/;
hiwjQ˛0 i D 0 ;
but this will be a higher order effect since w is small. One then proceeds by setting ˛ D 1;
M.u/ D M.Q/
and omits ˛ from the notation altogether. As before, one now decomposes everything in the form u D e i .Q C w/;
˝ ˛ ˙ D ˙ iwjG :
w D C GC C G C ;
(6.79)
The parameters governing the hyperbolic dynamics are 1 WD .C C /=2;
2 WD .C
/=2;
E WD .1 ; 2 / ;
whence w D 21 '
2i2
C :
The remainder satisfies the orthogonality condition ˝ ˛
jQ D
1
w 2 ; 2 2
˝ ˛ i jQ0 D 0;
hi jG˙ i D 0
which, by Lemma 6.17, is sufficient for the positivity property ˝
˛ 2 L j '
H 1 :
˝ ˛ i 0 0 The equation of is obtained by differentiating ˝ ˛ 0 D iuje Q D hiwjQ i. Using the equation of w (6.75), as well as w C 2Qjw D 0, one concludes that .P C 1/ M.Q/
˝
˛ ˝ wjQ0 D
˛ Lw C N.w/jQ0 D
˝ ˛ kwk22 C N.w/jQ0 :
6.3 The cubic radial NLS equation in R3
229
The equation for ˙ is obtained by differentiating (6.79). In fact, ˝ ˛ ˝ ˛ P ˙ D ˙ i wj P G D .P C 1/.Q C w/ Lw C N.w/j ˙ G D ˙˙ C N˙ .w/; N˙ .w/ WD hN.w/ C .P C 1/wj ˙ G i; E solves and so P 1 D 2 C N1 .w/; P 2 D 1 C N2 .w/;
˝ ˛ N1 .w/ D N.w/ C .P C 1/wji ; ˝ ˛ N2 .w/ D N.w/ C .P C 1/wj' :
Recall the energy expansion J.u/
J.Q/ D
˛ 1˝ L j C.w/ 2 1˝ ˛ 21 C L j C.w/: 2
C C
D 22
In analogy with Chapter 4 one therefore defines the linearized energy norm as ˝ ˛ ˝ ˛ E 2 C 1 L j D .2 C 2 / C 1 L j ' kvk2 1 ; kvk2E WD jj C H 2 2 2 where we used Lemma 6.17 for the final step. Furthermore, we define the smooth nonlinear distance function in such a way that, still under the mass constraint M.u/ D M.Q/, 2 dQ .u/ ' inf ku
2 e iˇ QkH 1
ˇ 2R
2 dQ .u/ D kwk2E
kwkE =.2ıE / C.w/ if dQ .u/ 1;
where ıE 1 is chosen such that kvkE 4ıE H) jC.v/j kvk2E =2: The smooth cut-off .r/ is equal to one on jrj 1 and vanishes for jrj 2. To see the consistency of the above two properties, let u D e iˇ Q C v be a minimizer for distH 1 .u; S1 / D inf ku ˇ
i
Then hwjiQ0 i D 0 implies that he kvkH
1
e iˇ QkH 1 :
vjiQ0 i D sin.ˇ
& inf jˇ k2Z
/M.Q/, and so
C kj
230
6 Further developments of the theory
as long as v is small. The case of k odd can be eliminated here via the sign in (6.78). Indeed, by the second condition in (6.78), cos.ˇ which excludes that ˇ
/.QjQ0 / > Re.vje i Q0 /
lies near an odd multiple of . Therefore, kwkE ' kwkH 1 . kvkH 1 kwkH 1 :
By the same argument, if u D e iˇ Q C v is an L2 distance minimizer, then jˇ
j . distL2 .u; S1 /;
provided that the right-hand side is small enough. In the region dQ .u/ 1, the distance function dQ enjoys the following properties: 2 kwk2E =2 dQ .u/ 2kwk2E ;
2 dQ .u/ D kwk2E C O.kwk3E /;
2 dQ .u/ ıE H) dQ .u/ D J.u/
J.Q/ C 221 :
Hence as long as dQ .u/ < ıE we have 2 @ t dQ .u/ D 41 P 1 D 42 1 2 C 41 N1 .w/:
In analogy with the results of Chapter 4 one develops eigenvalue dominance, and the ejection mechanism. We only reproduce the ejection lemma from [110], and refer the reader to that paper for everything else. Lemma 6.18. There exists a constant 0 < ıX ıE , as well as constants C ; T ' 1 with the following properties: Let u.t/ be a local solution of NLS in H1 on an interval Œ0; T satisfying R WD dQ u.0/ ıX ; J.u/ < J.Q/ C R2 =2 and for some t0 2 .0; T /, dQ u.t/ R .0 < 8t < t0 /: Then u extends as long as dQ u.t/ ıX , and satisfies 8 t 0 dQ u.t/ ' s1 .t/ ' sC .t/ ' e t R; j .t/j C k .t/kE . R C e 2t R2 ; sK u.t/ & .e t C /R;
6.3 The cubic radial NLS equation in R3
231
where s D C1 or s D 1 is constant. Moreover, dQ u.t/ is increasing for t T R, and jdQ u.t/ Rj . R3 for 0 t T R. As for NLKG, the key to characterizing long-term dynamics is to combine the ejection mechanism with the following variational lemma, which gives lower bounds on jKj. See Figure 4.3 for a schematic description of this lemma. For the definition of the functional I see (2.24). Lemma 6.19. For any ı > 0, there exist "0 .ı/; 0 ; 1 .ı/; 2 .ı/ > 0 such that the following hold: (i) For any u 2 H1 satisfying J.u/ < J.Q/ C "0 .ı/2 , and dQ .u/ ı, we have K.u/ 1 .ı/; or K.u/ min 1 .ı/; 0 kruk22 : (ii) For any u 2 H satisfying I.u/ < J.Q/
ı, we have
K.u/ min.2 .ı/; 0 kruk22 /: The one pass theorem. For the NLS equation, the one-pass theorem takes the following form: Theorem 6.20. There exist 0 < " R 1 with the following property: let u 2 C.Œ0; T /I H/ be a forward maximal solution of (6.65) satisfying M.u/ D M.Q/, J.u/ < J.Q/ C "2 and dQ u.0/ < R for some " 2 .0; " andR 2 .2"; R . Then one has the following either T D 1 and dQ u.t/ < R C R2 for all dichotomy: t 0, or dQ u.t/ R C R2 on t t < T for some finite t > 0. In the latter case, S.u.t// 2 f˙1g does not change on t t < T ; if it is 1, then T < 1, whereas if it is C1, then T D 1. In the Klein–Gordon case, we were able to use the same R for the dichotomy because the distance function was strictly convex in t. For NLS we do not obtain such a strong property, but rather need to allow for oscillations on the order of O.R3 /. Thus, we give ourselves some room (we chose R2 ) to ensure a true ejection from the small neighborhood. As before, the proof of this no-return statement hinges on a suitable virial identity. In fact, we use Ogawa–Tsutsumi’s saturated virial identity [113], (3.5) Z @ t hm ujiur i D 2jur j2 @r m dx juj2 .@r =2 C 1=r/m dx R3 Z (6.80) 4 juj .@r =2 C 1=r/m dx; R3
232
6 Further developments of the theory
where the smooth bounded radial function m is chosen as follows: m .r/ D m.r=m/;
0 0 m .0/ D 0 m .r/ 1 D m .0/;
00 m .r/ 0: (6.81)
Notice that with this choice of m , eq. (6.80) is not merely a cut-off of the virial identity, but rather a “smooth interpolate” of the latter with the Morawetz estimate for large jxj. This is indeed crucial for the following arguments, which are slightly more delicate than those in [113]. In analogy with the NLKG equation studied in the previous chapters, the idea is now to combine the hyperbolic structure of the ejection lemma close to the soliton manifold S with the variational structure in Lemma 6.19 away from S, in order to control this virial identity through the functional K.u/. We refer the reader to [110] for the details.
6.4
The energy critical wave equation
We conclude this chapter by presenting the results from [87] on the H 1 -critical, focusing nonlinear wave equation uR
u D juj2
2
u;
u.t; x/ W R1Cd ! R;
2 D
2d d
2
.d D 3 or 5/ (6.82)
in the radial context, where 2 denotes the H 1 Sobolev critical exponent. The dimensional restriction here is needed only for the blow-up characterization by DuyckaertsKenig–Merle [47]. We take the radial energy space as the phase space for the above equation, which can be normalized to L2 by setting uE WD .jrju; u/ P 2 L2radial .Rd /2 DW H p 1 at each time t 2 R, where jrj D is an isometry from HP radial .Rd / onto 2 d Lradial .R /. Thus, to any scalar space-time function u.t; x/, we will associate the vector function uE .t; x/ by the above relation. Conversely, for any time independent 'E D .'1 ; '2 / 2 H, we introduce the following notation ' WD jrj
1
'1 ;
'P WD '2 :
The conserved energy of (6.82) is denoted by Z h 2 juj P C jruj2 E.E u/ WD 2 Rd
juj2 i dx: 2
(6.83)
(6.84)
233
6.4 The energy critical wave equation
It is well-known that this problem admits the static Aubin–Talenti solutions which are of the form W D T W;
h W .x/ D 1 C
jxj2 i1 d.d 2/
d 2
;
(6.85)
where T denotes the HP 1 preserving dilation T '.x/ D d=2
1
'.x/ :
(6.86)
These are positive radial solutions of the static equation jW j2
W
2
W D 0;
(6.87)
which are unique, up to dilation and translation symmetries, amongst the nonnegative, non-zero (not necessarily radial) C 2 solutions, see [24]. They also minimize the static energy Z h i 1 1 2 j'j .x/ dx ; J.'/ WD jr'j2 (6.88) 2 Rd 2 among all non-trivial static solutions. The work of Kenig, Merle [84] and Duyckaerts, Merle [48] allows for a characterization of the global-in-time behavior of solutions with E.E u/ J.W /. In particular, these references construct the onedimensional stable and unstable manifolds associated with W . Note that [84] assumes the role for the energy critical equation which [114] and [77] played in the subcritical context. We wish to study the behavior of solutions with E.E u/ < J.W / C "20 ;
(6.89)
for some small "0 > 0. The key feature of (6.82) by contrast to NLKG is the scaling invariance of (6.82) manifested by d
u.t; x/ 7! 2
1
u.t; x/ D T u.t/
(6.90)
which leaves the energy unchanged. In particular, the analogue of the “one pass theorem” of Chapter 4 needs to be modified, specifically by replacing the discrete set of attractors fQ; Qg there by the one-parameter family of the ground states ˚ S WD ˙ WE >0 :
(6.91)
234
6 Further developments of the theory
Note that for the subcritical cubic NLS equation of [110], the scaling parameter (in frequency) is essentially fixed or at least bounded from above and below by the L2 conservation law, but for the energy critical equation there is no mechanism which a priori prevents the scale from going to 0 or C1. The “virial functional” as in (2.64) takes the form Z jr'j2 j'j2 dx K.'/ WD (6.92) Rd
and satisfies K.W / D 0. The following positivity is crucial for the variational structure around W H.'/ WD kr'k22 =d D J.'/
K.'/=2 :
(6.93)
Note that the derivative of J.'/ with respect to any scaling '.x/ 7! a '.b x/ except for T gives a non-zero constant multiple of K.'/. It is a special feature of the scaling invariant scenario that it suffices to work with a single K, whereas in the subcritical case we needed two different functionals and their equivalence. The main results from [87] can be summarized as follows. Theorem 6.21. There exist a small " > 0, a neighborhood B of SE within O." / distance in H, and a continuous functional ˚ S W 'E 2 H n B j E.'/ E < J.W / C "2 ! f˙1g; such that the following properties hold: For any solution u with E.E u/ < J.W / C "2 on the maximal existence interval I.u/, let ˚ I0 .u/ WD t 2 I.u/ j uE .t/ 2 B ; ˚ I˙ .u/ WD t 2 I.u/ j uE .t/ 62 B; S.E u.t// D ˙1 : Then I0 .u/ is an interval, IC .u/ consists of at most two infinite intervals, and I .u/ consists of at most two finite intervals. u.t/ scatters to 0 as t ! ˙1 if and only if ˙t 2 IC .u/ for large t > 0. Moreover, there is a uniform bound M < 1 such that 2.d C 1/ : d 2 For each 1 ; 2 2 f˙g, let A1 ;2 be the collection of initial data uE .0/ 2 H such that E.E u/ < J.W / C "2 , and for some T < 0 < TC , kukLqt;x .IC .u/Rd / M;
. 1; T / \ I.u/ I1 .u/;
q WD
.TC ; 1/ \ I.u/ I2 .u/:
Then each of the four sets A˙;˙ has nonempty interior, exhibiting all possible combinations of scattering to zero/finite time blowup as t ! ˙1, respectively.
6.4 The energy critical wave equation
235
The neighborhood (or “tube”) B as well as the sign functional S are defined explicitly. In short, every solution u with energy E.E u/ < J.W / C "2 can change its sign S uE .t/ at most once, namely by entering the neighborhood B, whereas u scatters/blows-up if it eventually maintains a fixed sign S D C1= 1. This is the same description as in the subcritical setting of the previous chapters, at least as far as the dynamics away from the ground states S is concerned. More specifically, the assertion about the sign change of the solution appears to be quite robust; it relies on the “one-pass” theorem, which in turn is based largely on energy and virial type arguments (recall that we needed Strichartz estimates to control the concentration of mass around zero energy, see Lemma 4.13, which required the equation to be energy subcritical; however, for the critical equation it turns that this particular issue can be dealt with differently). Unfortunately, it is presently unclear how to analyze solutions u which remain inside the tube B. Note that the blowup solutions constructed by Krieger, Tataru, and the second author [92] for the three-dimensional critical wave equation belong to this tube. We now recall the statement of that result. The local energy relative to the origin is defined as Z 2t C jrj2 C jj6 t; x dx : Eloc .E / D Œjxj 21 and ı > 0. Then there exists an energy solution u of (6.82) with d D 3 which blows up precisely at r D t D 0 and which has the following property: in the cone jxj D r t and for small times t the solution has the form, with .t/ D t 1 , 1 u.x; t/ D 2 .t/W .t/r C .x; t/ where Eloc E.; t/ ! 0 as t ! 0 and outside the cone u.x; t/ satisfies Z jru.x; t/j2 C ju t .x; t/j2 C ju.x; t/j6 dx < ı Œjxjt
for all sufficiently small t > 0. In particular, the energy of these blow-up solutions can be chosen arbitrarily close to E.W; 0/, i.e., the energy of the stationary solution. Duyckaerts, Kenig, and Merle [47] showed that all type-II blowup (i.e., blowup with uniformly bounded energy norm) under the constraint (6.89) is of the form described by Theorem 6.22, albeit without the explicit expression for .t/. But the tube around S might also contain solutions which do not blow up but rather scatter to S.
236
6 Further developments of the theory
This would correspond to the center-stable manifolds of Chapter 3. In contrast to the subcritical case, we do not address the existence problem of such a center-stable manifold associated with (6.82), nor do we give a complete description of all possible dynamics of solutions satisfying (6.89). Recall that [90] establishes the existence of such a manifold for the radial three-dimensional critical wave equation, but not in the energy class. It appears to be a delicate question in any dimension to decide whether or not a center-stable manifold associated with the ground states exists for energy critical equations. Note, however, that the one-dimensional stable (and therefore also unstable) manifolds were constructed in [48]. For the 4 C 1-dimensional energy critical wave equation, Hillairet and Raphaël [71] recently constructed C 1 type-II blowup solutions. As for subcritical equations, the proof of Theorem 6.21 relies on an interplay between the hyperbolic dynamics of the linearized operator around W with the variational structure of J and K away from S. The linearization around W is somewhat delicate as far as the dynamics is concerned. Indeed, one encounters a time-dependent scaling parameter .t/ which is associated with the zero mode of the linearized operator. This constitutes a major distinction from the analysis of the preceding chapters. Krieger and the authors [87] base their analysis of .t/ on the observation that the evolution of this parameter is much slower than that of the exponentially unstable mode. Indeed, the evolution of .t/ is governed by the threshold eigenvalue or resonance (which lies at zero energy) of the linearized operator and is therefore by nature algebraically unstable rather than exponentially unstable. This allows one to freeze the dilation parameter in those time intervals during which the trajectories are dominated by the hyperbolic (and unstable) dynamics. The other major difference, which could potentially be more serious, is the possibility of concentration blow-up in the region K 0 away from S. Recall that in the subcritical case we argued that all solutions are bounded in the phase space under these conditions and are therefore automatically global. Note that this issue arises after applying the one-pass theorem. Fortunately as well as crucially, the blowup analysis by Duyckaerts, Kenig, and Merle [47] precisely determines all possible blowup in that scenario, placing any blow solution of that type inside the tube B. Since we are dealing with solutions outside of this tube (for example, after ejection), one obtains the desired contradiction. It is not our intention to present an overview, let alone many details, of the precise arguments which [88] developed for the energy critical wave equation leading to the proof of Theorem 6.21. Rather, we content ourselves with a precise statement of the analogue of Lemma 4.8 as needed for that purpose. We remark that the proof presented below differs from the one which appears in [88].
237
6.4 The energy critical wave equation
6.4.1 The variational structure in the energy critical setting Define W to be the critical ground state in Rd , d 3, rescaled by > 0 so that krW k2 does not depend on . Further, Z h i 1 1 2 jruj2 juj dx; J.u/ D d 2 2 ZR (6.94) 2 2 jruj juj dx; K.u/ D Rd
as well as dQS .u/ D inf
Z
>0 Rd
ˇ ˇr.W
ˇ2 u/ˇ dx :
Then one has the following variational property outside of the soliton tube. Lemma 6.23. For each ı0 > 0 there is "1 D "1 .ı0 / > 0 sufficiently small such that if J.u/ < J.W / C "1 and dQS .u/ > ı0 , then we have either ˚ 2 K.u/ > min ı0 ; ckrukL 2 or else K.u/
0 and some absolute constant c > 0. Proof. By the critical Sobolev imbedding, the statement holds provided kruk2 < c0 where c0 > 0 is some absolute constant. Thus, assume the lemma fails and let P1 fun g1 nD1 H be a sequence with krun k2 > c0 ;
K.un / ! 0;
J.un / < J.W / C
1 : n
(6.95)
Since
1
run 2 C 1 K.un / (6.96) 2 d 2 is bounded in HP 1 \ L2 . By reflexivity of these spaces, J.un / D
we see that fun g1 nD1
run * ru1 ;
un * u1
(6.97)
weakly in L2 and L2 , respectively, as n ! 1 (henceforth, we will tacitly pass to subsequences). By (6.95) we can assume that
kun k22 ! c1 ;
krun k22 ! c1
(6.98)
238
6 Further developments of the theory
where c1 c0 . Since J.W / D d1 krW k22 , we infer from (6.95) and (6.96) that c1 krW k22 . Now assume the following:
(6.99)
ku1 k22 D c1
By the uniform convexity of the unit ball of L2 this is the same as un ! u1 strongly in L2 . We then conclude that u1 ¤ 0, that K.u1 / 0, and that J.u1 / J.W / as well as kru1 k2 krW k2 . If K.u1 / < 0, then K. u1 / D 0 with some 0 < < 1. However, kr. u1 /k2 < krW k2 which contradicts the variational characterization of W . Therefore, K.u1 / D 0 and kru1 k2 krW k2 implies that u1 must belong to the family of minimizers fW g>0 . Moreover, c1 kru1 k22 D krW k22 c1 whence we have krun k22 ! kru1 k22 ;
n!1
and therefore also run ! ru1 strongly in L2 . But this contradicts that dQS .u/ > ı0 . So it remains to prove (6.99). Via a sequence of dilations, we arrange that Z c1 8n jun j2 dx D (6.100) 2 Œjxj 0 and a sequence Rn ! 1 such that Z jun j2 dx > 8 n : (6.102) jxj>Rn
Let be a standard bump function which D 1 on jxj < 1, and define un D u1n C u2n D .x=n /un C 1 .x=n / un where n < Rn is chosen such that n ! 1 and krun k22 D kru1n k22 C kru2n k22 C o.1/ ;
kun k22 D ku1n k22 C ku2n k22 C o.1/ :
(6.103)
239
6.4 The energy critical wave equation
To accomplish this, one considers a dyadic partition of the set En;N D f2 N Rn < jxj < Rn g for N large and fixed, applied to Z jrun j2 C jxj 2 jun j2 C jun j2 dx (6.104) En;N
which is uniformly bounded in n. Making N D N."/ large, one can find a dyadic shell which makes the integral in (6.104) over that shell < ". This proves (6.103). By (6.102) and Sobolev imbedding, lim inf kru2n k22 > 0 n!1
and from (6.100), lim inf kru1n k22 > 0 : n!1
On the other hand, one has lim inf K.ujn / 0 n!1
for j D 1 or j D 2 (or both). But then we obtain a contradiction since lim sup krujn k22 < krW k22 ; n!1
j D 1; 2 :
Therefore, kk D c1 as desired. Next, suppose u1 D 0. First Z
jun j2 dx C
Z
jr.un /j2 dx
22
(6.105)
where C is the sharp constant in the Sobolev imbedding and is a test function. Passing to the limit here implies that Z
jj2 d C
22 jj2 d
Z
(6.106)
and thus .E/ C ..E// d
d 2
;
8 E Rd
(6.107)
D where E is Borel. It follows that , and D D 0 away from the atoms of , -a.e. Since only 0 can be an atom of by radiality, it follows that D c1 ı0 . However, this contradicts (6.100).
240
6 Further developments of the theory
Therefore, u1 ¤ 0. Set vn WD un
u1 . Then in the weak- sense of measures,
jrvn j2 *
jru1 j2 ;
(6.108)
jvn j2 *
ju1 j2 :
The latter statement here requires the following refinement of Fatou’s lemma: suppose fk 2 Lp is a bounded sequence with 1 p < 1. If fk ! f a.e. on a set U Rd , then p p p lim kfk kL kfk f kL p .U / p .U / D kf kLp .U / (6.109) k!1 Note that we may assume un ! u1 a.e. by the compact Sobolev embedding in Lp on bounded sets with 1 p < 2 . Applying the arguments of the paragraph beginning with (6.105) to vn implies that jru1 j2 C ˛ı0 ;
(6.110)
D ju1 j2 C ˇı0 where 0 ˇ C ˛ 2 2
C c1
2 2
. If ˇ > 0, then also ˛ > 0 and
i 22 hZ 22 d 2 C .R / jru1 j dx C ˛ C Rd Z 2 2 jru1 j2 dx 2 C ˛ 2 > C Rd Z ju1 j2 dx C ˇ D .Rd / D c1
(6.111)
Rd
2
so that C c12
1
> 1. However, by construction c1 krW k22 as well as
krW k22 D kW k22 D C krW k22 2
imply that 1 C c12
1
. This shows that ˇ D 0 and (6.99) holds.
Exercise 6.24. Prove (6.109). Hint: this is due to Brezis-Lieb, see [53], page 11.
References
[1] Agmon, S. Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. [2] Alinhac, S. Blowup for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications, 17. Birkhäuser Boston, Inc., Boston, MA, 1995. [3] Artbazar, G., Yajima, K. The Lp -continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Univ. Tokyo 7 (2000), no. 2, 221–240. [4] Bahouri, H., Gérard, P. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121 (1999), no. 1, 131–175. [5] Ball, J. Saddle point analysis for an ordinary differential equation in a Banach space, and an application to dynamic buckling of a beam, in Nonlinear Elasticity, (R. Dickey, ed.), Academic Press, New York, (1973), 93–160. [6] Bates, P. W., Jones, C. K. R. T. Invariant manifolds for semilinear partial differential equations. Dynamics reported, Vol. 2, 1–38, Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester, 1989. [7] Bates, P., Lu, K., Zeng, C. Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Amer. Math. Soc. 135 (1998), no. 645; Persistence of overflowing manifolds for semiflow. Comm. Pure Appl. Math. 52 (1999), no. 8, 983– 1046; Approximately invariant manifolds and global dynamics of spike states. Invent. Math. 174 (2008), no. 2, 355–433. [8] Beceanu, M. New estimates for a time-dependent Schrödinger equation, preprint 2009, to appear in Duke Math. J. [9] Beceanu, M. A critical centre-stable manifold for the Schroedinger equation in three dimensions, preprint 2009, to appear in Comm. Pure and Applied Math. [10] Beck, M., Wayne, C. E. Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws. J. Differential Equations 244 (2008), no. 1, 87–116; Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity. SIAM J. Appl. Dyn. Syst. 8 (2009), no. 3, 1043–1065. [11] Berestycki, H., Cazenave, T. Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 9, 489–492. [12] Berestycki, H., Lions, P.-L. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345; Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983), no. 4, 347–375. [13] Bergh, J., Löfström, J. Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
242
References
[14] Bianchini, S., Bressan, A. A center manifold technique for tracing viscous waves. Commun. Pure Appl. Anal. 1 (2002), no. 2, 161–190. [15] Bizo´n, P. Threshold behavior for nonlinear wave equations. Nonlinear evolution equations and dynamical systems (Kolimbary, 1999). J. Nonlinear Math. Phys. 8 (2001), suppl., 35–41. [16] Bizo´n, P., Chmaj, T., Tabor, Z. On blowup for semilinear wave equations with a focusing nonlinearity. Nonlinearity 17 (2004), no. 6, 2187–2201. [17] Bizo´n, P., Chmaj, T., Szpak, N. Dynamics near the threshold for blowup in the onedimensional focusing nonlinear Klein–Gordon equation, preprint 2010. [18] Bourgain, J. Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999. [19] Bourgain, J. Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc. 12 (1999), 145–171. [20] Bourgain, J., Wang, W. Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1–2, 197–215. [21] Brenner, P. On space-time means and everywhere defined scattering operators for nonlinear Klein–Gordon equations. Math. Z. 186 (1984), no. 3, 383–391; On scattering and everywhere defined scattering operators for nonlinear Klein–Gordon equations. J. Differential Equations 56 (1985), no. 3, 310–344. [22] Buslaev, V. S., Perelman, G. S. Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. (Russian) Algebra i Analiz 4 (1992), no. 6, 63–102; translation in St. Petersburg Math. J. 4 (1993), no. 6, 1111–1142. [23] Buslaev, V. S., Perelman, G. S. On the stability of solitary waves for nonlinear Schrödinger equations. Nonlinear evolution equations, 75–98, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995. [24] Caffarelli, L., Gidas, B., Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. [25] Carr, J. Applications of centre manifold theory. Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981. [26] Carr, J., Pego, R. L. Metastable patterns in solutions of u t D 2 uxx f .u/. Comm. Pure Appl. Math. 42 (1989), no. 5, 523–576; Invariant manifolds for metastable patterns in u t D 2 uxx f .u/. Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), no. 1–2, 133–160. [27] Cazenave, T. Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. [28] Cazenave, T., Lions, P.-L. Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982), no. 4, 549–561. [29] Chen, X., Hale, J. K., Tan, B. Invariant foliations for C 1 semigroups in Banach spaces. J. Differential Equations 139 (1997), no. 2, 283–318.
References
243
[30] Choptuik, M. W. Universality and scaling in gravitational collapse of a massless scalar field Phys. Rev. Lett. 70 (1993), 9–12. [31] Chow, S., Lu, K. Invariant manifolds for flows in Banach spaces. J. Differential Equations 74 (1988), no. 2, 285–317. [32] Chow, S., Lin, X., Lu, K. Smooth invariant foliations in infinite-dimensional spaces. J. Differential Equations 94 (1991), no. 2, 266–291. [33] Chow, S., Liu, W., Yi, Y. Center manifolds for smooth invariant manifolds. Trans. Amer. Math. Soc. 352 (2000), no. 11, 5179–5211. [34] Christ, M., Kiselev, A. Maximal functions associated to filtrations. J. Funct. Anal. 179 (2001), no. 2, 409–425. [35] Coffman, C. Uniqueness of the ground state solution for u u C u3 D 0 and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81–95. [36] Collet, P., Eckmann, J.-P. Extensive properties of the complex Ginzburg–Landau equation. Comm. Math. Phys. 200 (1999), no. 3, 699–722. [37] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3 . Ann. of Math. 167 (2008), 767–865. [38] Comech, A., Cuccagna, S., Pelinovsky, D. E. Nonlinear instability of a critical traveling wave in the generalized Korteweg-de Vries equation. SIAM J. Math. Anal. 39 (2007), no. 1, 1–33. [39] Constantin, P., Foias, C. Navier–Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. [40] Constantin, P., Foias, C., Nicolaenko, B., Temam, R. Integral manifolds and inertial manifolds for dissipative partial differential equations. Applied Mathematical Sciences, 70. Springer-Verlag, New York, 1989. [41] Costin, O., Huang, M., Schlag, W. On the spectral properties of L˙ in three dimensions. Preprint 2011. [42] Cuccagna, S. Stabilization of solutions to nonlinear Schrödinger equations. Comm. Pure Appl. Math. 54 (2001), no. 9, 1110–1145. [43] Cuccagna, S., Mizumachi, T. On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. Comm. Math. Phys. 284 (2008), no. 1, 51–77. [44] Demanet, L., Schlag, W. Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation. Nonlinearity 19 (2006), no. 4, 829–852. [45] Donninger, R., Schlag, W. Numerical Study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein–Gordon equation, preprint 2010, to appear in Nonlinearity. [46] Duyckaerts, T., Holmer, J., Roudenko, S. Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15 (2008), no. 6, 1233–1250. [47] Duyckaerts, T., Kenig, C., Merle, F. Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation, preprint 2009, to appear in
244
References J. Eur. Math. Soc. (JEMS); Universality of the blow-up profile for small type II blowup solutions of energy-critical wave equation: the non-radial case, preprint 2010, to appear in J. Eur. Math. Soc. (JEMS)
[48] Duyckaerts, T., Merle, F. Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18 (2009), no. 6, 1787–1840; Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP 2008 [49] Duyckaerts, T., Roudenko, S. Threshold solutions for the focusing 3D cubic Schrödinger equation. Rev. Mat. Iberoamericana 26 (2010), no. 1, 1–56. [50] Eckmann, J.-P., Wayne, C. E. Propagating fronts and the center manifold theorem. Comm. Math. Phys. 136 (1991), no. 2, 285–307. [51] Engel, K.-J., Nagel, R. A short course on operator semigroups, Universitext, Springer Verlag, 2006. [52] Erdogan, B., Schlag, W. Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. II. J. Anal. Math. 99 (2006), 199–248. [53] Evans, L. C. Weak convergence methods for nonlinear partial differential equations. CBMS Regional Conference Series in Mathematics, 74. AMS, 1990. [54] Fibich, G., Merle, F., Raphaël, P. Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation. Phys. D 220 (2006), no. 1, 1–13. [55] Flügge, S. Practical quantum mechanics. Reprinting in one volume of Vols. I, II. Springer-Verlag, New York–Heidelberg, 1974. [56] Gallay, T. A center-stable manifold theorem for differential equations in Banach spaces. Comm. Math. Phys. 152 (1993), no. 2, 249–268. [57] Gallay, T., Wayne, C. E. Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2 . Arch. Ration. Mech. Anal. 163 (2002), no. 3, 209–258; Global stability of vortex solutions of the two-dimensional NavierStokes equation. Comm. Math. Phys. 255 (2005), no. 1, 97–129. [58] Gesztesy, F., Jones, C. K. R. T., Latushkin, Y., Stanislavova, M. A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. Indiana Univ. Math. J. 49 (2000), no. 1, 221–243. [59] Gesztesy, F., Zinchenko, M. On spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279 (2006), no. 9-10, 1041–1082. [60] Gidas, B., Ni, Wei Ming, Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), no. 3, 209–243. [61] Ginibre, J., Velo, G. On a class of nonlinear Schrödinger equation. I. The Cauchy problems; II. Scattering theory, general case, J. Func. Anal. 32 (1979), 1–32, 33–71; Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401; The global Cauchy problem for the nonlinear Klein–Gordon equation. Math. Z. 189 (1985), no. 4, 487–505.; Time decay of finite energy solutions of the nonlinear Klein–Gordon and Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), no. 4, 399–442.
References
245
[62] Glassey, R. T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation, J. Math. Phys., 18, 1977, 9, 1794–1797. [63] Grillakis, M. Linearized instability for nonlinear Schrödinger and Klein–Gordon equations. Commun. Pure Appl. Math., 41 (1988), 747–774. [64] Grillakis, M. Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Commun. Pure Appl. Math., 43 (1990), 299–333. [65] Grillakis, M. Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math. (2) 132 (1990), no. 3, 485–509. [66] Grillakis, M. On nonlinear Schrödinger equations. Comm. PDE 25 (2000), 1827– 1844. [67] Grillakis, M., Shatah, J., Strauss, W. Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), no. 1, 160–197; Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94 (1990), no. 2, 308–348. [68] Guckenheimer, J., Holmes, P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990. [69] Hadamard, J. Sur l’Iteration et les Solutions Asymptotiques des Equations Differentielles, Bull. de la Soc. Math. de France 29 (1901), 224–228. [70] Hamza, M.-A., Zaag, H. A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case, preprint 2010. [71] Hillairet, M., Raphaël, P. Smooth type II blow up solutions to the four dimensional energy critical wave equation, preprint 2010. [72] Hirsch, M. W., Pugh, C. C., Shub, M. Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin–New York, 1977. [73] Hörmander, L. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Second edition. Springer Study Edition. Springer-Verlag, Berlin, 1990. [74] Holmer, J., Roudenko, S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm. Math. Phys. 282 (2008), no. 2, 435–467. [75] Howland, J. Perturbation of embedded eigenvalues. Bull. Amer. Math. Soc. 78, (1972) 380; Puiseux series for resonances at an embedded eigenvalue. Pac. J. Math. 55, 157 (1974). [76] Hundertmark, D., Lee, Y.-R. Exponential decay of eigenfunctions and generalized eigenfunctions of a non-self-adjoint matrix Schrödinger operator related to NLS. Bull. Lond. Math. Soc. 39 (2007), no. 5, 709–720. [77] Ibrahim, S., Masmoudi, N., Nakanishi, K. Scattering threshold for the focusing nonlinear Klein–Gordon equation, preprint 2010, To appear in Analysis and PDE. [78] Iooss, G., Vanderbauwhede, A. Center manifold theory in infinite dimensions. Dynamics reported: expositions in dynamical systems, 125–163, Dynam. Report. Expositions Dynam. Systems (N.S.), 1, Springer, Berlin, 1992.
246
References
[79] Jeanjean, L., Le Coz, S. Instability for standing waves of nonlinear Klein–Gordon equations via mountain-pass arguments. Trans. Amer. Math. Soc. 361 (2009), no. 10, 5401–5416. [80] Jörgens, K. Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77 (1961) 295–308. [81] Karageorgis, P., Strauss, W. Instability of steady states for nonlinear wave and heat equations. J. Differential Equations 241 (2007), no. 1, 184–205. [82] Keel, M., Tao, T. Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955–980. [83] Keller, C. Stable and unstable manifolds for the nonlinear wave equation with dissipation J. Diff. Eqs. 50 (1983), 330–347. [84] Kenig, C., Merle, F. Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166 (2006), no. 3, 645–675; Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201 (2008), no. 2, 147– 212. [85] Keraani, S. On the defect of compactness for the Strichartz estimates of the Schrödinger equation, J. Diff. Eq. 175 (2001), 353–392 [86] Killip, R., Vi¸san, M. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, preprint 2011. [87] Krieger, J., Nakanishi, K., Schlag, W. On the critical nonlinear wave equation above the ground state energy in R5 , to appear in American Journal Math. [88] Krieger, J., Nakanishi, K., Schlag, W. Global dynamics above the ground state energy for the one-dimensional NLKG equation, to appear in Math. Z. [89] Krieger, J.; Schlag, W. Non-generic blow-up solutions for the critical focusing NLS in 1-D. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 1, 1–125. [90] Krieger, J., Schlag, W. Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Amer. Math. Soc. 19 (2006), no. 4, 815– 920. [91] Krieger, J., Schlag, W. On the focusing critical semi-linear wave equation. Amer. J. Math. 129 (2007), no. 3, 843–913. [92] Krieger, J., Schlag, W., Tataru, D. Slow blow-up solutions for the H 1 .R3 / critical focusing semilinear wave equation. Duke Math. J. 147 (2009), no. 1, 1–53. [93] Kuroda, S. T. Scattering theory for differential operators, I and II, J. Math. Soc. Japan, 25 (1972), 75–104 and 222–234. [94] Levine, H. A. Instability and nonexistence of global solutions to nonlinear wave equations of the form P u t t D Au C F .u/. Trans. Amer. Math. Soc. 192 (1974), 1–21. [95] Li, Y., McLaughlin, D. W., Shatah, J., Wiggins, S. Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation. Comm. Pure Appl. Math. 49 (1996), no. 11, 1175–1255. [96] Li, C., Wiggins, S. Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations. Applied Mathematical Sciences, 128. Springer-Verlag, New York, 1997.
References
247
[97] Lieb, E., Loss, M. Analysis Graduate Studies in Mathematics, Vol. 14, AMS 1997. [98] Lions, P. L. The concentration-compactness principle in the calculus of variations. The locally compact case, part I Ann. IHP 1 (1984), 109–145; The concentrationcompactness principle in the calculus of variations. The limit case, part II Rev. Mat. Iberoamericana 1 (1985), 145–201. [99] Marzuola, J., Simpson, G. Spectral analysis for matrix Hamiltonian operators. Nonlinearity 24 (2011), no. 2, 389–429. [100] McLeod, K. Uniqueness of positive radial solutions of u C f .u/ D 0 in Rn . II. Trans. Amer. Math. Soc. 339 (1993), no. 2, 495–505. [101] Merle, F. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69 (1993), no. 2, 427–454. [102] Merle, F., Raphael, P. On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation. J. Amer. Math. Soc. 19 (2006), no. 1, 37–90; The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. (2) 161 (2005), no. 1, 157–222; On universality of blow-up profile for L2 critical nonlinear Schrödinger equation. Invent. Math. 156 (2004), no. 3, 565–672. [103] Merle, F., Raphael, P., Szeftel, J. Stable self-similar blow-up dynamics for slightly L2 super-critical NLS equations. Geom. Funct. Anal. 20 (2010), no. 4, 1028–1071. [104] Merle, F., Raphael, P., Szeftel, J. The instability of Bourgain–Wang solutions for the L2 critical NLS , preprint 2010. [105] Merle, F., Vega, L. Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D. Internat. Math. Res. Notices 1998, no. 8, 399–425. [106] Merle, F., Zaag, H. Determination of the blow-up rate for a critical semilinear wave equation. Math. Ann. 331 (2005), no. 2, 395–416; Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math. 125 (2003), no. 5, 1147–1164. [107] Mizumachi, T. Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 48 (2008), no. 3, 471–497. [108] Morawetz, C. S. Strauss, W. A. Decay and scattering of solutions of a nonlinear relativistic wave equation. Comm. Pure Appl. Math. 25 (1972), 1–31. [109] Nakanishi, K., Schlag, W. Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. Journal Diff. Eq. 250 (2011), 2299–2333. [110] Nakanishi, K., Schlag, W. Global dynamics above the ground state energy for the cubic NLS equation in R3 . Preprint 2010. To appear in Calc. Var. PDE [111] Nakanishi, K., Schlag, W. Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation without the radial assumption. To appear in Arch. Ration. Mech. Anal. [112] Nakanishi, K., Schlag, W. Center-stable manifolds around soliton manifolds for the nonlinear Klein–Gordon equation. Preprint 2011. [113] Ogawa, T., Tsutsumi, Y. Blow-Up of H 1 solution for the Nonlinear Schrödinger Equation, J. Diff. Eq. 92 (1991), 317–330.
248
References
[114] Payne, L. E., Sattinger, D. H. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22 (1975), no. 3–4, 273–303. [115] Pecher, H. Low energy scattering for nonlinear Klein–Gordon equations. J. Funct. Anal. 63 (1985), no. 1, 101–122. [116] Perelman, G. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2 (2001), no. 4, 605–673. [117] Pillet, C. A., Wayne, C. E. Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Diff. Eq. 141 (1997), no. 2, 310–326. [118] Promislow, K. A renormalization method for modulational stability of quasi-steady patterns in dispersive systems. SIAM J. Math. Anal. 33 (2002), no. 6, 1455–1482. [119] Raphaël, P. Stability and Blowup for the nonlinear Schrödinger equation, Lecture notes, Zürich 2008. [120] Reed, M., Simon, B. Methods of modern mathematical physics, Academic Press, London, 1979. [121] Schlag, W. Harmonic analysis notes, www.math.uchicago.edu/schlag [122] Schlag, W. Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. Ann. of Math. (2) 169 (2009), no. 1, 139–227. [123] Schlag, W. Dispersive estimates for Schrödinger operators: a survey. Mathematical aspects of nonlinear dispersive equations, 255–285, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007. [124] Schlag, W. Spectral theory and nonlinear partial differential equations: a survey. Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 703–723. [125] Shatah, J. Unstable ground state of nonlinear Klein–Gordon equations. Trans. Amer. Math. Soc. 290 (1985), no. 2, 701–710. [126] Shatah, J., Struwe, M. Geometric wave equations. Courant Lecture Notes in Mathematics, 2. American Mathematical Society, Providence, RI, 1998. [127] Shatah, J. Zeng, C. Orbits homoclinic to centre manifolds of conservative PDEs. Nonlinearity 16 (2003), no. 2, 591–614. [128] Soffer, A., Weinstein, M. Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133 (1990), 119–146; Multichannel nonlinear scattering, II. The case of anisotropic potentials and data. J. Diff. Eq. 98 (1992), 376–390. [129] Soffer, A., Weinstein, M. I. Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16 (2004), no. 8, 977–1071. [130] Stanislavova, M., Stefanov, A. On precise center stable manifold theorems for certain reaction-diffusion and Klein–Gordon equations. Phys. D 238 (2009), no. 23-24, 2298– 2307. [131] Strauss, W. A. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), no. 2, 149–162. [132] Strauss, W. A. Nonlinear invariant wave equations. Invariant wave equations (Proc. “Ettore Majorana” Internat. School of Math. Phys., Erice, 1977), 197–249, Lecture Notes in Phys., 73, Springer, Berlin–New York, 1978.
References
249
[133] Strauss, W. A. Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. [134] Struwe, M. Globally regular solutions to the u5 Klein–Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 495–513. [135] Sulem, C., Sulem, P-L. The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. [136] Tao, T. Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. AMS Providence, RI, 2006. [137] Tsai, T. P., Yau, H. T. Stable directions for excited states of nonlinear Schroedinger equations, Comm. Partial Differential Equations 27 (2002), no. 11&12, 2363–2402. [138] Vanderbauwhede, A. Centre manifolds, normal forms and elementary bifurcations. Dynamics reported, Vol. 2, 89–169, Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester, 1989. [139] Wayne, C. E. Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch. Rational Mech. Anal. 138 (1997), no. 3, 279–306. [140] Wayne, C. E. Vortices and two-dimensional fluid flow, Notices AMS, 2011. [141] Weder, R. The Wk;p -continuity of the Schrödinger wave operators on the line. Comm. Math. Phys. 208 (1999), no. 2, 507–520. [142] Weder, R. Center manifold for nonintegrable nonlinear Schrödinger equations on the line. Comm. Math. Phys. 215 (2000), no. 2, 343–356. [143] Weinstein, M. I. Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. [144] Weinstein, M. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. [145] K. Yajima, The W k;p -continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47 (1995), no. 3, 551–581; The Lp boundedness of wave operators for Schrödinger operators with threshold singularities. I. The odd dimensional case. J. Math. Sci. Univ. Tokyo 13 (2006), no. 1, 43–93; The Lp boundedness of wave operators for Schrödinger operators with threshold singularities. I. The odd dimensional case. J. Math. Sci. Univ. Tokyo 13 (2006), no. 1, 43–93. [146] Zeng, C. Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Comm. Pure Appl. Math. 53 (2000), no. 10, 1222–1283; Erratum: Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Comm. Pure Appl. Math. 53 (2000), no. 12, 1603–1605.
Index
1-parameter groups of symmetries, 30 9-set theorem, see nine set theorem almost homoclinic orbits, 168 asymptotic profiles, 19, 38, 39 Aubin–Talenti solutions, 233 Bahouri–Gérard decomposition, 11, 19, 38, 179 Beceanu’s dispersive estimate, 122, 203 Bernstein’s inequality, 73 Besov spaces Bp;2 , 64, 73 Birman–Schwinger operator, 120 blowup after ejection, 177 center manifold theorem, 5, 90 center/stable/unstable manifolds, 4, 85 Christ–Kiselev lemma, 75 completely integrable, 1, 14 complex formalism, 197, 201 computer experiments, 8 concentration-compactness, 4, 11, 19, 38–43, 47–56, see also Bahouri–Gérard decomposition conservation laws, 2 conservation of free energy, 25 conserved energy, 2, 17 critical element, 37, 50, 56, 181 cutoff w.t; x/, 165 C.v/, 151
embedded eigenvalues, 83 energy critical wave equation, 232 energy estimate, 20 energy landscape, 80–84 energy splitting, 116 energy-subcritical regime, 1, 2, 11, 12, 17 essential spectrum, 124, 125 Euler–Lagrange equation, 31, 33, 46 exercises (star, dagger), 12 explicit soliton, 6, 204 exponential weight, 114 failure of compactness, 39 finite speed of propagation, 18, 38 finite time blowup, 18, 25 focusing, 2 forbidden trajectory, 156 forward scattering region, 10, 27 fractional integration, 71 Fredholm operator, 198 Frobenius theory, 26 G0 functional, 32 Galilei transforms, 196 gap property, 11, 80, 84, 116, 173 global NLKG dynamics, 173 global-in-time dynamics, 189 graph transform, 10, 79, 108, 115, 143, 176, 187, 222 ground state, 3, 29
defocusing, 2 dispersion relation, 64 dispersive Hamiltonian equations, 12 distorted Fourier transform, 209–213 Duhamel formula, 20
Hamiltonian formulation, 2, 24, 103, 198 Hamiltonian vector field, 198 Hartman–Grobman theorem, 85 Herglotz function, 210 homoclinic orbit, 163
eigenmode dominance, 145, 153, 201, 230 ejection lemma, 153, 161, 171, 188, 201, 230 ejection process, 145, 146, 153, 230
induction on energy, 178 infinite dimensional hyperbolic dynamics, 14 instability of the ground state, 31, 35, 81, 122
252
Index
intertwining property, 120 invariant cones, 11, 79 J functional, 3, 29–36 K0 functional, 3, 30–36, 81, 145–148, 158–162 K2 functional, 46, 51, 149, 158–159 K˛;ˇ functionals, 45 Keel–Tao endpoint, 76 Kenig–Merle method, 11, 19, 37, 174, 178 Klein–Gordon (NLKG) equation, 17 Lagrange multiplier, 33 Lagrangian, 1 Laurent expansion, 118 limit-point case, 210 limiting profiles, see asymptotic profiles linear dispersive estimates, 63–77, 117–129 linear profile decomposition, 42, 43, 179, see also Bahouri–Gérard decomposition linearized energy, 151, 229 linearized NLKG, 102 linearized operator LC , 80 Littlewood–Paley decomposition, 54, 64, 70, 72, 73 local conservation law, 61 local energy, 235 local energy density, 61 local well-posedness theory, 17 Lorentz contraction, 195 Lorentz transform, 1, 39, 53, 58, 104, 194 Lorentz transformed wave equations, 60 Lp bound on the wave operators, 120 Lumer–Phillips theorem, 105 Lyapunov–Perron method, 10–14, 76, 79, 80, 108–117, 130–143, 173, 188, 189, 207, 222, 226 minimal energy, 194 modulation of parameters, 11, 76, 123, 130–143, 194–203, 227 momentum, 2, 53, 57 Morawetz estimate, 37, 232 negative energy, 35
negative spectrum, 9 negatively invariant, 88, see also positively invariant nine set (or 9-set) theorem, 6, 12, 173, 175, 191–194, 206, 220, 227 nodal solutions, 29 Noether’s theorem, 2 noncompact group of symmetries, 39 nondegenerate Hessian, 66 nonlinear distance function, 145, 151, 201 nonlinear profile, 42, 47, 48, 180 nonlinear profile decomposition, 43, 48, 178–180 nonlinear Schrödinger equation, 122, 218 nonradial cubic NLKG, 194–203 nonradial scattering, 52 (non)stationary phase, 65 numerical evidence, 5, 8 ODE-type blowup, 25–27, 206 one-dimensional KG equation, 6, 7, 75, 204–218 one-parameter family of ground states, 233 one-pass (or no-return) theorem, 8, 12, 146, 162–171, 174, 189, 191, 201, 233 orthogonality condition, 132, 135, 137, 141, 200, 228 orthogonality of the free energy, 40, 54, 55 Payne–Sattinger criterion, 19 Payne–Sattinger region, 9 Payne–Sattinger regions, 3, 34 Payne–Sattinger theorem, 34–36, 188 Payne–Sattinger well, 31 perturbation lemma, 43 perturbation theory, 19, 43 perturbative analysis near Q, 188 Plancherel identity, 211 Pöschl–Teller potentials, 209 pointwise decay, 42, 64, 119 positively (negatively) invariant, 88, 92, 93 radial cubic NLS in R3 , 130–143, 218–232 regular/singular threshold, 117 resonance, 117 resonance function, 118
Index returning trajectories, 167 Riesz projections, 107, 110, 134 rigidity argument, 19, 51, 57, 179 root space, 124, 134, 206, 227 saddle surface, 9, 34, 80, 81 saturated virial identity, 231 scattering after ejection, 177 scattering to the ground states, 80, 173, 189 Schwarz symmetrization, 32, 34 sign of K0 and K2 , 46, 160 sign S.E u/, 162 singular continuous spectrum, 120 singular thresholds, 117, 121 small data scattering, 19, 21, 49, 60, 188 small Strichartz norm, 43, 169 smooth codimension 1 manifold, 9, 137, 196, 222 Sobolev critical exponent, 232 soliton tube, 235, 237 space-time translations, 39, 53 spectral measure, 118, 211–215 spectrum of the linearized operator, 9, 79, 108, 125, 206 speed of light, 3 stable blowup regime, 13 stable/unstable manifold, 4–14, 81, 85, 88, 89, 106, 116, 122, 147, 173, 188, 189, 206, 223 stationary energy, 3, 11, 18, 29, 188, see also J functional stationary phase method, 65–69 stationary solution, 4, 11, 19, 20, 29–34, 77, 108, 191, 193 Stone’s formula, 211 Strauss’ estimate, 29, 207 Strichartz estimates for KG, 63–77
253
Strichartz estimates for linearized NLS, 125 Sturm oscillation theory, 83, 207 symmetric nonincreasing rearrangement, 34 symplectic decomposition, 198, 200, 227 symplectic form, 2, 198, 226 threshold behavior (Duyckaerts–Merle theory), 13 threshold resonance, 116, 121, 192, 197, 206, 208 threshold solutions, 6, 13, 189, 197, 205 trapped by the ground states, 8, 156–158, 173–175, 186, 193–196, 222 trapped trajectory, 157 traveling waves, 3, 104, 195, 201 T T -method, 70 type-II blowup solutions, 193, 235, 236 unitary correspondence, 213 unstable ground state, see instability of the ground state unstable hyperbolic dynamics, 11 unstable manifold, see stable/unstable manifold vanishing kinetic energy, 169 virial identity, 8, 20, 51, 52, 164, 165, 179, 193, 221, 231–234 Volterra equation, 83, 208, 214 wave operators, 28, 112, 119–121, 218 Weyl criterion, 106 Weyl theory, 210 Weyl–Titchmarsh functions m˙;˛ , 212 Weyl–Titchmarsh solutions, 212 Yajima’s theorem, 112, 119–121