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0. We show the effect of the strict local maxima and minima of Q(x) on the existence of multi-peaked positive solutions as h becomes small. Using variational methods, we construct positive solutions condensing at the extremal points of Q(x). An explicit expression for the dominant parts of the solutions is also obtained.
1
Introduction and Statement of the Main Result
The aim of this paper is to establish the existence of single and multi-peaked positive solutions to the following problem -h2Au + u = Q(x)up-1, lim u{x) = 0,
xeKN,
(1.1) (1.2)
|x|—>oo
where A is the Laplace operator, Q(x) is a positive function of locally Holder continuous on RN(N > 1), 1 < p < $±§ if TV > 3; 1 < p < +oo if TV = 1, 2, and h > 0 is a constant. The existence of positive solutions of (1.1)-(1.2) and its various generaliza tions have been studied by many authors in the past twenty years. In addition to the papers mentioned below, the readers may consult the survey articles [9,11] and the references cited therein. Most of them provide the existence of solutions for arbitrary h > 0. When Q(x) has one or more nondegenerate critical points, Cao, Noussair, and Yan [5] established single-peaked and/or multi-peaked positive solutions concentrating near a neighborhood of those points. In [4], Cao and Noussair obtained the existence of k positive solutions with energy close to that of leastenergy solutions, when Q(x) has k global maxima and ft > 0 is sufficiently small.
16
It is then natural to ask whether we can still establish the existence of single-peaked and/or multi-peaked positive solutions, when Q(x) has one or more local maxima or minima which may be degenerate critical points. Similar problem was considered by Gui in [8]. Following the methods originally intro duced by Sere [17] and Coti-Zelati and Rabinowitz [7], Gui used the method of minimaxing the penalized functionals for a special family of "mountain passes" in [8] and obtained the existence of multi-bump solutions for the problems sim ilar to (1.1)-(1.2). The existence of periodic solutions was also considered by Alama and Li [1] using similar arguments of [17,7]. The purpose of this paper is to answer the question raised above. To state our result more precisely, some preparations are in order. First we recall some known facts about positive solutions to the problem -Au + u = up~\ xeRN, lim u(x) = 0, u(0) = maxxeKNu(x).
(1.3) (1.4)
|x|-»oo
It is proved that the problem (1.3)-(1.4) has a unique positive solution [/(ground state) satisfying that U is spherically symmetric and its first deriva tives decay exponentially. Next we introduce some notations. For U , I J G i7 1 (R i V ), let (u, v)h = / (h2 Vu\7v + uv),
(1.5)
INlX = /(ft 2 |Vu| 2 + |ti|2).
(1.6)
Denote r m m = {a € R N
Fmax = {a e K
| a is a strict local minima
ofQ(x)},
| a is a strict local maxima
ofQ(x)}.
Define, for a,y G R ^ ,
Eh,a,y = {ue &{*") I (u,u(x~l~y))h {U
'
dXi
= o,
)^ = 0,i = l,.--,N}.
The main result in this paper is the following.
(1.7)
17
Theorem 1.1. Suppose there exist k distinct points a 1 , • • • , afc such that {a , • • • ,ak} C Tmin or {a 1 , • • • ,a fc } C Tmax. Then, for h small enough, (1.1)(1.2) has a positive solution Uh of the form 1
uh
= J2 0 , t = l , " - , t , ( j / l e rii=i^M',j/i» ajl->(Q(ai))"^2,
(1.8)
'
|i£|—>0,
5WC
^ ^ a * ' ^ o r ^ "^ °'
\\u;h\\h = o(h%).
Our strategy in proving Theorem 1.1 is to use a type of Lyapunov-Schmidt reduction to reduce the problem we are dealing with to a finite dimensional one and then solve the finite dimensional problem by considering an auxilinary minimization or maximization problem. Our method is a combination of the variant of those used in [2,15] and energy comparison used in [12,13]. Throughtout this paper all integrals are over R ^ . We shall use the same letter C to denote various generic positive constants and use 0{t) to mean \0(t)\ < C\t\. o{\) will denote various quantities that tend to zero as h —> 0. 2
Proof of the Main Result
We shall only give the proof of Theorem 1.1 for the case k = 2 since the case k = 1 is similar but simpler and the case k > 3 can be proved similarly.By change of variables x —> x/h we see that Uh is a solution of (1.1) (1.2) if and only if Vh = Uh(x/h) is a solution of -Au + u = Qh\u\p~2u, lim u(x) = 0,
xeRN,
(2.1) (2.2)
|sc|—>oo
where Qh(%) — Q(hx). Corresponding to (•, -)h,Eh,a,y defined in Section 1, we define < u, v >= / (VuVv + uv), and let ||.|| be the usual norm introduced by (•,•). Fh,aty = {ue HX{RN)
| (u,Uhiaiy)
= 0,
( t i , ^ ^ > = 0 , » = l,-,JV}>
(2.3)
18
where Uh^y = U(x - S±*). Theorem 1.1 is equivalent to the following result. Theorem 2.1. Under the same assumptions of Theorem has a positive solution Vhof the form
Vh
1.'1,(2.1)(2.2)
(x)=£aiUh9aiiyih+uh
(2.4)
i=l
with alh > 0, ylh G R
, i = 1, • • •, k, Uh G f]i=1 Fhaiyi
satisfying as h —> 0
ai->(Q(a'))-^,
(2.5)
\V\\ "► 0,
(2.6)
IKII -»• 0.
(2.7)
Let a 1 , a 2 be two points in Tmin or r m a x , cr0 = (§fet)
P
and £ > 0 be a
number to be determined. Denote by Br the open ball in R ^ of radius r > 0 centered at the origin. Our arguments are based on establishing the existence of critical points of the functional Kh : H1(RN) \—> R defined by
KM
_ J(iv~l' + lf) (JQM'V
via seeking critical points of Jh{yl,y2,cr,uj)
= Kh(Uhjaijyi
+ (cr0 + o-)Uh,a2,y2 + ^)
(2.9)
in the manifold M* = {
(z/1,?/2,^,^) | (y\y2,a) 2
G A^,
2=1
where TV* = {(z/1,2/2, cr)!^^ e Bs,i = l , 2 ; a G R , | a | < 6} The one to one correspondence of critical points of Jh in M$ and Kh in jff 1 (R JV ) when h and £ are small enough has been established in [5].We note that (y1 ,y2 ,a,u) G Ms is a critical point of Jh if and only if there exist
19
(ai, a 2 ) G R 2 , 0% = (/?j, • • •, /3^) G R N , 2 = 1,2 satisfying the following ' ^
= 0,
(*)c
TV
(*) { 2=1
2 = 1 Z = l
y i
Similar to [5],our approach in solving (*) is a combination of the implicit function theorem method used in [2],[16] and the energy comparison method used in [12,13].We first solve (*)u; and (*)a for each fixed pair (y 1 ,?/ 2 ) G B2§ x B25 by arguments used in [16] and then solve (*)yi via considering the following minimization problem (if a 1 , a 2 G Tmin) (**) \ni{Jh{y\y\a{y\y2),u{y\y2))
I (vl,V2)
eBsx
Bs }
or via considering the following maximization problem (if a1, a2 G Tmax) (* * *) s u p i J ^ y 1 ^ 2 , ^ 1 ^ 2 ) , ^ 1 ^ 2 ) ) | (y\y2)
eB6x
B6}
where a(yx,y2),uj(yl,y2) are determined by (*) -F/i.a1,!/1 f)Fh,a2,y2
Such t h a t
A
Ql
yi,y2(u)
= (Alyl^u,u)
for all u e Fh^aijyi f|^,a 2 ,t/ 2 Define Ah^yiy2 : Fh^y\^y2 \—> Fh^yi^y2
(2.40)
as follows
2
^,yi,l/ W= W l y i ^ ^ i j / i ^ w + ^ y y )
(2.41)
for w = (
^,^), 0,yiheKN,Cjhe
Fhjaiiyih f]Fh,a^yl
a{ -> Q~i±*{a%
\y\\ -> 0,
satisfying for i = 1,2 \\uh\\ -> 0,
(2.58)
as h —>> 0. The positivity of ?i^ can be obtained the same as in [5]. Thus Theorem 2.1 is proved. Acknowlegment. visiting the Institute of a research fellow of the foundation for partially
Part of this work was carried out while the author was Mathematics of the University of Mainz, Germany as Alexander von Humboldt Foundation. He thanks the financial support of this work.
References 1. S. Alama and Y. Y.Li, On "multibump" bounded states for certain semilinear elliptic equations, Indiana Univ. Math. J. 41(1992), 983-1025. 2. A. Bahri and J. Coron, On a nonlinear elliptic equation involving critical Sobolev exponents, Comm. Pure Appl. Math. 41(1988), 225-294. 3. A. Bahri, Y. Li, and O.Rey, On a variational problem with lack of com pactness: the topological effect of the critical points at infinity, Cal. Var. 3(1995), 67-93. 4. D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in R N , Ann. Inst. H. P. Anal. Non lineaire, 13(1996), 557-588.
27
5. D. Cao, E. S. Noussair and S. Yan, Solutions with multiple "peaks" for nonlinear elliptic equations, to appear in Proc. Royal Soc. Edinburgh, Sect.A. 6. D. Cao, N. Dancer, E. S. Noussair and S.Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete and Continuous Dynamic Systems, 2(1996), 221-236. 7. V. Coti-Zelati and P. Rabinowitz, Homoclinic type solution for a semilinear elliptic PDE on R N , Comm. Pure Appl. Math. 45(1992),12171269. 8. C. Gui, Existence of multi-bump solutions for Schrodinger equations via variational method, Comm. PDE. 21(1996), 787-820. 9. P. L. Lions, On positive solutions of semilinear elliptic equations in un bounded domains, In: Nonlinear Diffusion Equations and Their Equilib rium States, Ni, W.-M., Peletier, L. A., Serrin, J. (eds.), New York,Berlin: Springer 1988. 10. M. K. Kwong, Uniqueness of positive solutions of Au — u -f up — 0 in Rn, Arch. Rat. Mech. Anal. 105(1989), 243-266. 11. W.-M. Ni, Some aspects of semilinear elliptic equations. Lecture notes, National Tsing Hua University, Taiwan, 1988. 12. W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44(1991), 819-851. 13. W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70(1993), 247-281. 14. W.-M. Ni and J. Wei, On the location and profile of spike-layer solu tions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math.48(1995), 731-768. 15. P. Rabinowitz, On a class of nonlinear Schrodinger equations, Z. Angew. Math. Phys. 43(1992), 270-291. 16. 0 . Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89(1990), 1-52. 17. E. Sere, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209(1992), 27-42.
28
LARGE-TIME B E H A V I O R OF E N T R O P Y SOLUTIONS IN V FOR MULTIDIMENSIONAL CONSERVATION LAWS GUI-QIANG CHEN Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730, USA E-mail: [email protected] HERMANO FRID Instituto de Matemdtica, Universidade Federal do Rio de Janeiro C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil E-mail: [email protected] Dedicated to Professor Xiaqi Ding on the Occasion of His 70th Birthday A b s t r a c t . We study the large-time behavior of entropy solutions in L°° of the Cauchy problem for inviscid scalar conservation laws in any space dimension, using the approaches developed in [2,3,4]. For the initial data that are periodic in each space variable, we show that the corresponding periodic solution u(t, x) decays to the average of the initial data over the period in Lj , oc (M n ), as t —> oo, provided that the linearly degenerate set of the nonlinear flux function has measure zero. For the initial data that are any L1 nL°° perturbation of Riemann data determining a piecewise Lipschitz Riemann solution R(£), we prove that the solution u(t, t£), con sidered as a function of the variables t and £ = x/t, converges to R(£) in Lfoc(Rn), as t —> oo. These results significantly extend the known asymptotic results, which mostly apply only for the one dimensional case, to the invicid multidimensional case.
1
Introduction
In this paper we are concerned with the large-time behavior of entropy solutions in L°° of the Cauchy problem for a scalar conservation law in several space TTO •pi O V) 1 QC •
f dtu + 6ivxf(u) \u\t=0=u0(x),
= 0,
x € Rn, * > 0,
(1) )
where u G l and f(u) = (fi(u), • • •, fn(u)) is a vector function in C3(M; Rn). We consider two distinct cases: the initial data that are periodic in each space variable and the ones that are L1 C\ L°° perturbation of Riemann data. For the first case, we show that the solution u(t, x) decays to the average of the
29 initial data over the period in Lf oc (M n ), as t -» oo, for 1 < p < oo, provided that the set of linear degeneracy of the nonlinear flux function has measure zero (see (8)). In the second case, we prove that the solution u(t,t£), considered as a function of the variables t and £ = x/t, converges to the corresponding Riemann solution R(£) in Lf o c (E n ), as t -» oo, for 1 < p < oo, provided that the Riemann solution is piecewise Lipschitz in £. A pair (n(u),q(u)), q — (g 1 ,.••> qn), is called an entropy-entropy flux pair for (1) if 7] and q are Lipschitz continuous and satisfy q'(u) = r,'(u)f'(u).
(2)
Definition. A bounded measurable function, u{t,x), is called an entropy solu tion of (1) in UT = W1 x (0,T) if, for any entropy-entropy flux pair (77, g) of (1) with convex 77(14), /
{r)(u)(j)t + q(u) • V x 0 } dxdt + /
n(uo(x))(j)(0, x) dx > 0,
(3)
for any nonnegative function (f>(t,x) 6 Co(M n + 1 ). The existence of global entropy solutions of (1) with u0 G Loc(Wl) was first proved by Kruzkov [20], by improving an earlier result of Volpert [33] for UQ G BV{Wl). The large-time behavior of entropy solutions for the one dimensional case has been extensively studied. For example, see [8,15,18,22,26] and the references cited therein and in [14,16,19,27, 29,35]. In this paper, we apply our new approaches, developed in Chen-Frid [2,3,4], to deal with the asymptotic problems for the inviscid equations (1) in any space dimension for periodic initial data and general L1 C\ L°° initial perturbations of Riemann data. In Section 2, we show how the large-time decay of L°° periodic entropy solutions of (1) is achieved in any space dimension. Here we give a direct simpler proof of the compactness result of Lions, Perthame, and Tadmor [25], which allows the application of Theorem 2.1, leading to the decay result of Theorem 2.3. The decay of entropy solutions for scalar conservation laws was also established by Lax [21] and Dafermos [7] for the one-dimensional case and by Engquist-E [11] for the two-dimensional case with periodic initial data of locally bounded variation. In Section 3, we study the large-time behavior of entropy solutions of (1), when no is a general L1 D L°° perturbation of Riemann initial data. The convergence of any entropy solution u(t, ££) to the corresponding Riemann so lution R(£) was established in the sense of time-average, as t —» 00, in [3] for general initial data. For initial data that are an L 1 D L°° initial perturbation of Riemann data producing a planar Riemann solution, the convergence of the
30
corresponding entropy solution to the Riemann solution was also proved in Lf 0C (R n ), 1 < p < oo, as t —> oo in [3]. Here we extend these results to general initial data that are L1 fl L°° initial perturbation of Riemann data determining piecewise Lipschitz Riemann solutions. We remark the generality of such Rie mann data which generate piecewise Lipschitz Riemann solutions. Efforts in studying the piecewise Lipschitz structure of multidimensional Riemann solu tions for (1) have been made in many recent publications (cf. [1,6, 24,34,36]). 2
Large-Time Decay of Periodic Solutions
In this section we study the large-time decay of periodic solutions of multidi mensional scalar conservation laws, without local BV restriction on the L°° periodic initial data UQ G L°°(Rn) with period P — IIf =1 [0,p;]: u0(x+piei)
- u0(x),
z = l,---,n.
(4)
We first recall the basic theorem of [2] (see [3] for a detailed proof). This theorem holds not only for scalar equations but also systems of conservation laws. Let u(t,x) be a periodic solution of (1) as a hyperbolic system so that u € E m and f(u) e (M m ) n . We define the scaling sequence uT, T > 0, associated with u(t, x) by uT(t,x)
=u(Tt,Tx).
(5)
Theorem 2.1. Let the system be endowed with a strictly convex entropy. As sume that u(t,x) £ L°°(M" +1 ) is its periodic solution, with u0 G L°°(Rn) satisfying (4), and that the associated scaling sequence uT(t,x) is compact in Llc(Mn++1). Then esslim / \u(t,x) — u\pdx — 0,
for any
1 < p < oo,
(6)
t-»oo Jp
where u is given by 1
r
\p\ JP fp
uo(x)dx.
(7)
We apply Theorem 2.1 to analyze the large-time behavior of periodic solutions. First, we investigate the L\oc compactness of the entropy solu tion operator of (1). In this connection, we prove the following theorem first established by Lions, Perthame, and Tadmor in [25]. Here we give a direct proof of this compactness result, motivated by their ideas. Denote a(v) = (ai(v),---,a"(v)) = (f{(v),---, / » ) .
31
Theorem 2.2. Assume that, for any (r, jfc) G E x W1, with r 2 + \k\2 = 1, meas { v G E | r + a(v) •fc= 0 } = 0.
(8)
Then the solution operator u(t,-) = StUo(-) : L°° —> L°°, determined by (1), is compact in L}oc{(0,T) x E n ) . Proof. Let u(t,x) be any entropy solution of (1) in the sense of (3). Set f{t,x,v)
= Xu(t,x)(y),
where X\(v)
= H(v)H(X -v)-
H(-v)H(v
- A),
and W
\1,
S>1,
is the Heaviside function. We first prove that such a function f(t,x,v)
satisfies
dtf(t, x, v) -f a(v) - Vzf (t, x, v) — dvm(t, x, v) in the sense of distributions in (0,T) x E n x E for certain m(t,x,v) that m(t,x,v)
(9) satisfying
is a nonnegative Radon measure in (0,T) x E n x E.
(10)
Specifically, by entropy inequality (3), we deduce dt(u(t,x)
- v)+ +divxH(u(t,x)
- v)(f(u(t,x))
- f(v)) - - m i ( t , r r , v ) , (11)
and dt(v -u(t,x))+
+divxH(v
-u(t,x))(f(v)
- f(u(t,x)))
= -m2(t,x,v),
(12)
in the sense of distributions in (0, T) x E n xE, where mi and rri2 are nonnegative Radon measures (the Schwartz lemma [30]). Here we use the notation (5)+ = sH(s), s G E. Now we proceed formally (and we will justify our approach later) in the following way. We take the derivative of (11)-(12) with respect to v to find dtH(u(t, x) — v) + a(v) - VxH(u(t, x) — v) = dvm\ (t, x, v),
(13)
dtH(v — u(t,x)) + a(v) • VxH(v - u(t,x)) = —dvm2(t,x,v).
(14)
and
32
We then multiply (13) by H(v) and (14) by H(-v), to obtain dtf + a(y) • Va-f = dvm(t,x,v)
- (mi(t,x,0)
and take their difference
-m2(t,x,0))
®5v=:o,
(15)
where Sv=0 is the usual Dirac measure concentrated at v = 0 and m(t, x, v) = H{v)m\ (t, x, v) + H(—v)rri2(t, x, v).
(16)
Now we have mi(*,x,0) = ft (*(*,*))+ + div x #(iz(t,x))(/(u(*,:r)) - /(0)), m 2 (t,a;,0) = ft(-ti(t,x))+ + divxH{-u(t,x))(f(0) f(u(t,x))), which implies mi(t,x,0)
— ra2(£,x,0) = dtu(t,x)
+ div x /(i/(^,x)) = 0.
Therefore, we have proved (9) at least formally. Now we verify that all our formal procedures can be rigorously justified with the usual arguments in the theory of distributions. The only part that requires more explanation is the use of the formula H(v)dvmi
(t, x, v) = dvH(v)mi
(t, x, v) — mi (t, x, 0) 5v=o
in the sense of distributions and the analogous one for rri2(t,x,v) and H(—v). We notice that the measures m* may be written into sliced form as m,i(t, x, v) = filv(t,x) dv1 i = 1,2, where, for each v G M, \xxv is the Radon measure in (0,T) x Mn, which is given by the same expression defining m*, when v is viewed as a parameter for i = 1,2. Therefore, we can easily justify the above formula using a mollifying sequence Hs(v) for H(y) and then taking limit in the sense of distributions when 6 —> 0. The details can easily be filled out. Given a sequence of initial data UQ(X),SUP£>0 \\UQ\\L < oo, there exist uniform bounded solutions u£(t,x) of (1). It suffices for Theorem 2.2 to show that ue(t,x) is compact in Zj oc ((0,T) x Rn). Now, for each u£(t,x), we associate the function f£(t,X,v)
=Xu*(t,x)(V)>
which satisfies (9) in the sense of distributions in ( 0 , T ) x l n x E, for a nonnegative Radon measure m£(t,x,v) obtained from (11), (12), and (16) by sub stituting u(t,x) by u£(t,x). Note that / X\(v)i/>(v) dv = / JK
JO
ip(v)dv,
33
for any function ijj G L / 1 oc (E u ). In particular, u£{t,x)
= / f£(t,x,v)dv.
(17)
Therefore, we need to prove that / f£(t,x,v)dv
is compact in L}oc((0,T)
x En).
(18)
We notice that f£(t,x,v) = 0, if \v\ > Ro, for any R0 > 0 larger than the uniform bound of u£{t,x) in L°°((0,T) x E n ) . In particular, the integral in (18) may be taken over a fixed finite interval (—RQ,RQ). Now, from (11), (12), and (16), we see that the measures m£ have uniformly bounded total variation over any compact subset of (0, T) x E n x E. This follows from the uniform boundedness of mf and m | in Mioc((0, T)xRn xK), which, in turn, follows from the proof of the Schwartz lemma [30]. Hence, by the Sobolev embedding, we see that m£ form a compact set in Wz~c1,p((0,T) x E n x E) with 1 < p < g±f- (cf. [12]). Also, by (11), (12), and (16), we see that m£ are uniformly bounded in W~ 1 , o o ((0,T) x E n x E). Hence, using the interpolation arguments (see [28]), we obtain that m£ form a compact set in Wi; c l j 2 ((0,T)xlir x E ) . Next, we localize our problem in the following way. Given any compact set in (0,T) x E n , we multiply (9) by a C°° function that identically equals one over this compact set and has compact support contained in (0,T) x E n . For simplicity, we keep the notation f£ as the product of the original f£ by this test function. We then see that the new functions f£ satisfy an equation like (9) with m£ belonging to a compact set in W /r_1 ' 2 ((0,T) x E n x E) and have supports contained in a fixed compact set in (0, T) x E n x E. We may then write dvm£(t,x,v)
= (I - d2v)(I -
At,xy/2g£■
Then f£ satisfy dtf(t, x, v) + a(v) • V*f(*, x, v) = (I-
82V)(I - At,x)1/2&,
(19)
with g = g e belonging to a compact set in L 2 ((0,T) x E n x E). Let g£l be a subsequence of g£ converging in L 2 to certain g G £ 2 ((0, T) x E n x E). We may assume, passing to a subsequence if necessary, that f£l converges weak-star to certain function f G L°°((0,T) x E n x E) with compact support contained in (0,T) x E n x ( - i ? 0 , # o ) . Clearly, f and g satisfy (19) as well.
34
Now we prove that / f£l(t,x,v)dv
-> / f(t,x,v)dv,
JR.
in
L 2 ((0,T) x E n ) .
(20)
JH
Denote ~ the Fourier transform in (t,x).
It suffices to prove that
f F~i(T,k,v)ip{v)dv->
ff(T,k,v)il;(v)dv,
Jn
JR.
in L 2 (E x E n ) ,
(21)
for any \j) G CQ°(—RI,RI), with i?i > i?o- By Plancherel's identity, one has (20) while taking ip G CQ°(—RI, RI) that identically equals one over (—R0, Ro). Let f(t,x,v), g(t,x,v) G L 2 ( E x E n x E) satisfy (19). We take C € C£°(E) so that ( = 1 in (—1,1) and £ = 0 outside the interval (-2,2). Let S > 0 denote a number to be suitably chosen later. As in [10], we write /
fy(v)dv
= I1+I2,
(22)
JR
where
(23) For (r, k) G E x E n , with r 2 + |&|2 = 1, let //(£) be the distribution function li(t) = fiT>k(t) = m e a s { v G ( - i ? i , # i ) | |r + a(v) • A;| < * } , * > 0. Applying the Cauchy-Schwarz inequality to Ii, we obtain that, for r 2 4\k\ > 0, 2
|f|2dv)
\h\ 0.
Now, for r 2 + \k\2 > 7 2 , we choose 5 = S'(r2 + I&I2)1/2, where 8' > 0 is a small number. We then get \h\ 1. We then have the following immediate corollary of Theorem 2.2. Corollary 2 . 1 . Assume that UQ satisfies the periodic condition (4) and that condition (8) holds. Let u(t,x) be the entropy solution of (1) in E ^ + 1 . Set uT(t,x) — u(Tt,Tx). Then the self-similar scaling sequence uT is compact in Llc((0,oo)xW). Proof. Since u(t,x) e L°°(IR^ +1 ), then I|UT||LOO < C < oo,
where C is independent of T. Since u(t,x) is the periodic entropy solution, it satisfies dtri(u) + Vx • q(u) < 0 in the sense of distributions, for any convex entropy-entropy flux pair (rj^q). Thus uT also satisfy dtv{uT) + V x • q{uT) < 0, which implies that uT(t, x) is a sequence of entropy solutions of (1) in [0, oo) x Rn with oscillatory initial data u0(Tx). Theorem 2.2 implies the result we expected. □ Corollary 2.1 together with Theorem 2.1 yields the main result of this section. T h e o r e m 2.3. Let u(t,x) be the entropy solution of (1) in [0, oo) x Rn with uo satisfying (4)- Assume that condition (8) holds. Then esslim \\u(t, •) -
U\\LP(P)
where 1 < p < oo and u — rpr Jp UQ(X) dx.
= 0>
(32)
38
This can be seen as follows. From Theorem 2.1, it suffices to show that the corresponding self-similar scaling sequence uT(t,x) = u(Tt,Tx) is compact in L L ( M + + 1 ) - T h i s directly follows from Corollary 2.1. 3
Large-Time Behavior of General Entropy Solutions in L°°
We are now concerned with the large-time behavior of entropy solutions of the initial value problem for scalar conservation laws (1), with the initial data u0 G L°°(Rn) satisfying u0(x) = Ro(x/\x\) + Po(x), for a.e. a ; E l n , R0 G L ^ S " " 1 ) , P0 G L1 H L°°(M n ). Denote by R(x/t)
,^\ (M)
the unique entropy solution of (1) with initial data: u\t=o = R(x/\x\),
and call it the Riemann convergence u(t,tt;) —> average. Theorem 3.1. [4]. Let Riemann solution of (1)
(34)
solution of (1) and (34). In [4], we established the i?(£)> m ^ / / 1 o c (^ n )' as t -> oo, in the sense of timeu be the entropy solution of (1). Let R(x/t) be the and (34)- Assume uo G L°° satisfies (33). Then
lim ^ / \u{t,tO-R(0\dt T->oo 1 J0
= 0,
fora.e.^eW1.
(35)
We now prove that the convergence in the sense of (35) actually implies the convergence in the usual sense as t goes to infinity. This is given by the following result, which is motivated from Theorem 2.1 and [2,31]. Theorem 3.2. Let u(t,x) be the entropy solution of (1). Let R(£), £ = x/t, be the Riemann solution of (1) and (34)- Assume R(£) is piecewise Lipschitz continuous in the variable £, in the sense that R G jBV/0C(Mn); the closure of the jump set has measure zero, and outside the null set the first-order derivatives of R(£) are uniformly bounded. Then, for any 1 < p < oo, esslim / \u(t, t£) - R(0\pd£
= 0,
for any 17 <e
(36)
Proof. Let (ry(ix), q(u)) be a strictly convex entropy pair of (1). Denote (a(u, i>), P(u,v)) a family of entropy pairs, parametrized by v and formed by the quadratic parts of n and q at v: a(u,v) — 7](u) — r)(v) — Vr)(v)(u — v), 0{u,v) = q(u) - q{v) - Vn(v)(f(u) - f(v)).
39 Since u is an L°° entropy solution of (1), one has dtniu) + V* • q(u) < 0
(37)
in the sense of distributions. Let J be the closure of the jump set of R(£), which is a null set in the £-space by assumption, and B be any open ball in Rn — J . For (£, x) in the cone {(t, x) | x/t E B}, one has dtR + Vx-f{R) = 0,
(38)
dtr](R) + V* • q(R) = 0.
(39)
Then we obtain dta(u,R)
+ V x • /?(ii,i?) < -\/2r](R)(VxR,Qf(u,R))
(40)
in the sense of distributions, where Qf(u,v) = /(it) - /(i>) - Vf(v)(u — v) is the quadratic part of / at u. Now, since u is just an L°° function, we consider a mollifying kernel u £ Q>°(-1,1), CJ > 0, fRu(t)dt = 1, and set c^(*) = u(t/6)/6, 6 > 0. We will use the notation h6 = h*ws, for any function /i depending on t. Then, from (40), we get dta6(u,R)
+ Vx.(3d(u,R)
< - {\72r)(R)(VxR,Qf(u,R)))d
.
(41)
We now use the change of coordinates (t,x) *-> (£,£), € = xl^- Inequality (41) then becomes dta5(u,R)-^^as(u,R)+-tVc0s(u,R)
< - QvVW{12,Q/(u,.R))) .
(42) The derivatives with respect to £ in (42) should be taken in the sense of distributions (except those applied directly to R). We consider a nonnegative smooth function of £, 0 £ CQ°(B), such that (£) = 1, in {£ E i? | dist(£, c?I?) > e } , e > 0 sufficiently small. Applying (42) to the test function 0(£) yields
It*
-
t
'
for some constant C > 0, where we denote
Y$(t)= [ as(u(t),R)oo T J0
(44)
= 0.
We will prove that esslim Y(t) = 0.
(45)
t—KX)
Indeed, we have (* - f ) ^ ( ' ) 2 = 2 Jjs
- | ) ( F / ) ' ( S ) r / ( S ) ds + ^
Y«(s)2 da,
and thus use (43) to get ^Y*(T)2
< Cj^
^Yfit)
dt + J^ Y*{t? dt.
(46)
Now, in the above inequality, we can make —> 1 in B, keeping \\(/>\\oo and Var{0} bounded, and then make 5 -> 0 to get Y(T)2 +oo. Hence (36) is completely proved. □ Asymptotic Problems and Multidimensional Riemann Problems. The assumption that the Riemann solution i?(f) is piecewise Lipschitz in f in The orem 3.2 is quite general. This assumption is related to the structure problem of multidimensional Riemann solutions. The structure problem has been ex tensively studied in recent years for multidimensional scalar conservation laws. See [1,6,17,24,34] and the references cited therein for two dimensional Rie mann problems. In Zhang-Zheng [36], piecewise Lipschitz solutions were constructed under the assumption that f"(u) ^ 0, j — 1,2, and (fi/fy'Y ^ 0 for the Riemann problem that the space axes are all initial discontinuities and the initial value in the four quadrants are different constants. In Chen-Li-Tan [6], the Riemann problem, whose data are three constants in three fan domains forming different angles, was considered, and the dependence of the piecewise structure of the solutions upon the value of the constants as well as the angles was studied. All the solutions of these Riemann problems are piecewise Lipschitz and, therefore, these Riemann solutions are always asymptotically stable under L1 DL°° initial perturbation with the aid of Theorem 3.2. It would be interesting to study further the piecewise Lipschitz structure of solutions of multidimensional Riemann problems.
All the re
sults established in this paper hold for the viscous servation laws in several space variables by using the arguments developed in Chen-Frid [4]. Acknowledgments: Gui-Qiang Chen's research was supported in part by the National Science Foundation grants DMS-9623203 and DMS-9708261, and by an Alfred P. Sloan Foundation Fellowship. Hermano Frid's research was supported in part by CNPq-Brazil, proc. 352871/96-2. REFERENCES
1. Chang, T. and Hsiao, L., The Riemann Problem and Interaction of Waves in Gas Dynamics, Longman Scientific and Technical (Pitman Mono graphs No. 41), Essex, 1989. 2. Chen, G.-Q. and Frid, H., Asymptotic decay of solutions of conservation laws, C. R. Acad. Sci. Paris, Serie I, 323 (1996), 257-262. 3. Chen, G.-Q. and Frid, H., Decay of entropy solutions of nonlinear con servation laws, Arch. Rat. Mech. Anal. 1998 (to appear). 4. Chen, G.-Q. and Frid, H., Large-time behavior of entropy solutions of conservation laws, J. Diff. Eqs. 1998 (to appear). 5. Chen, G.-Q. and Frid, H., Divergence-measure fields and conservation laws, submitted (1997). 6. Chen, G.-Q., Li, D., and Tan, D.-C, Structure of the Riemann solutions for two-dimensional scalar conservation laws, J. Diff. Eqs. 127 (1996), 124-147. 7. Dafermos, C M . , Applications of the invariance principle for compact processes. II. Asymptotic behavior of solutions of a hyperbolic conserva tion law, J. Diff. Eqs. 11 (1972), 416-424. 8. Dafermos, C. M., Regularity and large time behavior of solutions of a conservation law without convexity, Proc. Royal Soc. Edinburgh 99A (1985), 201-239. 9. Ding, X. and Liu, T.-P., Nonlinear Evolutionary Partial Differential Equations, Proceedings of the International Conference held in Beijing, June 21-25, 1993, AMS/IP Studies in Advanced Mathematics 3, Amer. Math. Soc. Providence, RI; International Press, Cambridge, MA, 1997. 10. DiPerna, R. J. and Lions, P. L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729-757. 11. Engquist, B. and E, W., Large time behavior and homogenization of solutions of two-dimensional conservation laws, Comm. Pure Appl. Math. 46 (1993), 1-26.
43
12. Evans, L. C , Weak Convergence Methods for Nonlinear Partial Differ ential Equations, CBMS 72, AMS, Providence, 1990. 13. Evans, L. C. and Gariepy, R. F., Lecture Notes on Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, Florida, 1992. 14. Freisttiler, H. and Serre, D., L1 stability of shock waves in scalar viscous conservation laws, Comm. Pure Appl. Math. 51 (1998), 291-301. 15. Glimm, J. and Lax, P., Decay of solutions of systems of nonlinear hy perbolic conservation laws, Amer. Math. Soc. Memoir 101, A.M.S.: Providence, 1970. 16. Goodman, J., Stability of .viscous scalar shock fronts in several dimen sions, Trans. Amer. Math. Soc. 311 (1989), 683-695. 17. Guckenheimer, J., Shock and rarefactions in two space dimensions, Arch. Rat. Mech. Anal. 59 (1975), 281-291. 18. Il'in, A. M. and Oleinik, O. A., Asymptotic behavior of solutions of the Cauchy problem for some quasilinear equations for large values of time, Mat. Sbornik 51 (1960), 191-216. 19. Jones, C. K. P. T., Garner, R., and Kapitula, T., Stability of travelling waves for nonconvex scalar viscous conservation laws, Comm. Pure Appl. Math. 46 (1993), 505-526. 20. Kruzkov, S. N., First-order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243. 21. Lax, P., Hyperbolic systems of conservation laws, Comm. Pure Appl. Math. 10 (1957), 537-566. 22. Lax, P., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS. 11, SIAM, 1973. 23. Li, B.-H. and Wang, J.-H., The global qualitative study of solutions of a conservation law (I),(II), Sci. Sinica, 1979, Special Issue on Math. I (1979), 12-24, 25-38 (Chinese. English summary). 24. Lindquist, W. B., Scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown, SIAM J. Math. Anal. 17 (1986), 1178-1197. 25. Lions, P. L., Perthame, B., and Tadmor, E., A kinetic formulation of mul tidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), 169-192. 26. Liu, T.-P., Invariants and asymptotic behavior of solutions of a conser vation law, Proc. Amer. Math. Soc. 71 (1978), 227-231. 27. Matsumura, A. and Nishihara, K., Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity Comm. Pure Appl. Math. 165 (1994), 83-96. 28. Murat, F., L'injection du cone positif de H~l dans W~l,q est compacte
44
pour tout q(x) defined in x > 0, such that u(x,y) is of Cp when y ^ (x). Besides, u(x,y) satisfies the following estimates: < C 0 (x 3 + 2/ 2 ) 1/6 = «(°) in n 0 U Sl+
( ) ductively: F{ V)
s^ec^n-
(0,0))
\s(-,/'>(x,y)-s^(x,y)\ 0(b < 0) can leave ft± only at the side x = 0. The fact allows us to use characteristic method to estimate s^ + 1 ^ and its derivatives. Lemma 5.3 Under the assumptions of validity of
F^
|(*- aM)dysM
,, K
M, LV
°-
Integrating along the characteristics we have \v(x,y)\ < I* \(-\{;} + \f Using the assumptions F[v , F 2
+/»
-(T _ „ ( o ) ) ^ s « » - dyX^v
+ dy(\{;} -
\f)dys^ (5.19)
By integrating we obtain v(a, b) = f [ ( - A f + A f + a(#> - 4°)) + (A,» - r*"" 1 '! + \H(rM,sM)
-
ff^^s*"-1')!
< C|r, s("») - ff(r(">, s^" 1 *)] - [ff (0, «) - H(0,8^-^)\ = (\x'+(so)
< \\f"(m0)+O([m})\ + 0(\s^
• | ^ | • |*(") -
s ^ \
- aW|))|5))|s - r ^ - ^ I U - + \\X'+(s0)\
s^'l
\\sM - 8 - ^ - ^ J f l y ^ M
Integrating along the characteristics yields |v| < / 0 a | ( ^ - 1 ) - A ^ + a - A ^ ) ^ - 1 ) -a^)dys(°HaM^a,b)))\ da (6.4) In (6.5) the estimate of dys{-°\dy{s^ - S - 1 } . Notice that A; = o(di) implies Vi(0)->oo while di = o(ai) indicates that the images of ql are pushed to infinity. Hence the sequence {v^ is blowing up at one and only one point in the half space H. First we will show that
^ ( ^ ( 0 ) - { ^ -
R T
^ +^ }
do)
w i t h a ^ l i m t ^ ^ ] 1 1 ^ and e = ((),••-,0,2). Then we will apply Pohozaev identities in a large half ball in H to derive a contradiction. Step 1. Deriving (10) . First, by Lemma 2.3, for any r < 1, Vi(x)vi(0) < - ^ - , Vx G Br(0).
(11)
One may regard Vi as the solution of the linear problem J — Avi = Ci(x)vi x G Di i Vi(x) =0 x G dD.
(12)
Applying the well-known Harnack inequality (see [5]), we conclude that Vi satisfies (11) in any bounded set in Di and it follows that Vi(x)vi(0)-+w(x)
:= p r ^ + h(x) xeH \x\
(13)
hat Vi(x) — 0 on dDi, we write
where h(x) is some harmonic function. Due to the fact t
74
Here b{x) is another harmonic function on H with zero boundary value on dH. By the wellknown classification result on harmonic functions on half spaces, we must have b(x) = c(xn + 1) (14) for some constant c. More precisely, we must also have c — ^ r - Otherwise, h(0) ^ 0, and we can apply the similar argument as in [15] to derive a contradiction as follows: Apply the Pohozaev identity (6) to V{ on a small ball Bp(0), and multiply both sides by v2(0). Then the left hand side tends to 0, while the right hand side converges to _( n ~ 2 ) \Sn~l\h(0). This is impossible. To summerize, we have
~a
h{x)= v ;
2+%±i>.
\x + e\n~2
(15)
2n~2
v
}
Step 2. Applying the Pohozaev identities to derive a contradiction. Now we use the Pohozaev identities to V{ on G*, the intersection of some large ball Br(0) with D{. First, Vi satisfies -Avi
= RidiX+p^vJix)
x e d.
(16)
If we apply (6) to v\ and multiply both sides by v2 (0), then the left hand side becomes II = divUO) I x • \/R{d{x +pi)vT+1, (17) JGi
while by (13), the right hand side converges to the integral _
f r 2n (dw 1 . l2x / ^ ( ^ ~ x ' V^ - ox " v V^ ) + JdGn-2 vv 2 We prove the following IR
Claim 1.
=
IL~+0,
as
dw. nw-^-]. ov
(18)
i^oo.
Claim 2. IR becomes very negative as r gets large. Obviously, these two claims contradict with each other. Proof of Claim 1. Case (a). S7R{p°) ^ 0. First apply the second Pohozaev identity (7) to vi on B1/2(0) and multiply both sides by v2(Q). By (13), the right hand side is bounded. And consequently, by (8), there is some constant C, such that div2{0) < C
(19)
75
On the other hand, through a straight forward calculation, one can easily verify that
" MV+1~7f,
(20)
/.
where D is any bounded domain in Di and 7* = [v{(0)] n~2. Now Claim 1 follows from (19), (20) and the boundedness of \/R. Case(b). V#(p°) = 0 . In this case, the 'flatness condition' (i?4) applies. Let x1 — pl — p°. If \xl\ = 0(7i), then the 'flatness condition', the bound edness of \/R, a n d (20) imply immediately that
\IL\ < CdivU0h?^0. Notice here a > n — 2. Therefore, we may assume that there is a sequence of numbers {K{}, with \x{\ > Km
and K^oo.
(21)
On one hand, applying the second Pohozaev identity (7) and by a straight forward calculation, one can verify that divlmxY-1
It follows that Is—> — 00 as r—>oo. This completes the proof of Claim 2 and thus derives a contradiction in Case(ii).
77
2.3
Eliminating case (iii) .
Let Vi(x) — \
n-2 2
Ui(XiX + pl)
Lemma 2.1 guarantees that Vi converges in #1/2(0) with Vi(0) = 1. This in fact implies the following Claim 3. vi converges weakly in H1 to a function v in some half space H := {x = (xi, • • •, xn) E Rn : xn > c] satisfying
{
-Av = R(p°)vT t/(0) = 1
xeH (31)
v(x) = 0 xedH. However, due to the wellknown classification result, problem (31) has only trivial solution. This is a contradiction. Now what left is to justify Claim 3. Suppose V{ does not converge weakly, then it would blow up at some point zi with d(zl,dDi)—>0. Let Z{ be the preimage of z\. Let r; = d(zl, dft). Make a rescaling n-2
Wi(x) - ri
2
Uifcx + z1).
Then W{ has only one possible blow up point in H. Similar arguments as in Case (i) and (ii) will lead to a contradiction.
3
Completing t h e Proof of Theorem 1
So far we have established (5), which indicates that Si are discrete. Further more, we can show that Si are bounded away from dft. In fact, suppose in the contrary, there exists pl E Si with d(pl, d£l)-*0. Now there are only two possi bilities: Xi = o(di) or di = O(Xi). One can use an entirely similar argument as in subsection 2.2 or 2.3 to derive contradictions. Therefore, the solutions are uniformly bounded in a neighborhood of dft. It follows that the solutions {ui} can only blow up at finitely many points in the interior of f£^~. Assume that
are those isolated blow up points. Again we will use the Pohozaev identity to derive a contradiction. First apply (6) to equation (1) in fi \ (Be(pl), • • • ,Be{pm)) for some small e and multiply both sides by uf(pu). By Lemma 2.4, we have
ii,(p»)«i(*H«,(*) :=
|g_%|n.a
+ • • • + | g _ ^ | , - 8 + M*)
(32)
78
with some harmonic function h(x) in ft. It follows that -™ E
f
r
2n ,dw
1
.
l9,
dw.
f
:r 2 [^^(^7 -V^-^^-HV^| ) + ^ —V ] = / du l
"=i /
k n 2
JdBe{p ) - '
°
n
.
l2
2 -—-x-i/lvH n Z
JdQ -
(33) where z/ is the unit outward normal vector. Since ft is star-shaped, the right hand side is positive. Hence at least one of the integrals on the left is positive, say f
. 2n ,dw
/
h—o^x
1
-^w-ox-
. v
l9x
dw,
„
\VH ) + nwar] > °-
/rtJX
(34)
On the other hand, applying (6) on Be(p1), we find that the left hand side u2i(pU) [
x-vRuT+i-tO,
however (34) infers that the right hand side must be positive. A contradiction. Therefore, {ui} are uniformly bounded in ft. This completes the proof of Theorem 1.
References [1] S. Alama, G. Tarantello, On semilinear elliptic equations with indefi nite nonlinearities, Calculus of Variations and PDE, 1(1993), 439-475. [2] A. Bahri, J. Coron, The scalar curvature problem on three-dimensional sphere, J. Funct. Anal., 95(1991) 106-172. [3] H. Berestycki, I. C. Dolcetta, L. Nirenberg, Superlinear indefinite el liptic problems and nonlinear Liouville theorems, Topological methods in Nonlinear Analysis, 4(1994), 59-78. [4] H. Berestycki, I. C. Dolcetta, L. Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, to appear. [5] L. Caffarelli, G. Fabes, E. Mortola, S. Salsa, Boundary behavior of nonnegative solution of elliptic operator in divergent form, Indiana Univ. Math. J. 30(1981), 621-640. [6] W. Chen, C. Li, A priori estimates for prescribing scalar curvature equations, Annals of Math., 145(1997), 547-564. [7] W. Chen, C. Li,
Prescribing scalar curvature on 5 3 ,
preprint, 1996.
79 [8] W. Chen, C. Li, A note on Kazdan-Warner type conditions, Geom. Vol. 41, No. 2(1995) 259-268.
J. Diff.
[9] W. Chen, C. Li A priori estimates for solutions to nonlinear elliptic equations, Arch. Rat. Mech. Anal., 122(1993) 145-157. 10] W. Chen, C. Li Indefinite elliptic problems in a domain, and Continuous Dynamical Systems, 3 (1997), 333-340.
Discrete
11] A. Chang, P. Yang, A perturbation result in prescribing scalar curva ture on 5 n , Duke Math. J., 64(1991) 27-69. 12] A. Chang, M. Gursky, P. Yang, The scalar curvature equation on 2and 3-spheres, Calc. Var., 1(1993) 205-229. 13] J. Escobar, R. Schoen, Conformal metric with prescribed scalar cur vature, Invent. Math. 86(1986) 243-254. [14] Z. H. Han, Y. Y. Li, The Yamabe problem on manifolds with bound ary: existence and compactness results, preprint. [15] Y. Y. Li, Prescribing scalar curvature on Sn and related problems, J. Differential Equations 120 (1995), 319-410. [16] Y. Y. Li, The Nirenberg problem in a domain with boundary, Topological methods in nonlinear analysis 6(1995), 309-329. [17] Y. Y. Li, Prescribing scalar curvature on Sn and related problems, Part II: Existence and compactness, Comm. Pure and Appl. Math. 49 (1996) 541-597. [18] P. Rabinowitz, Minimax methods in critical point theory with appli cations to differential equations, CBMS Lecture No. 65, 1986. [19] R. Schoen, D. Zhang, Prescribed scalar curvature on the n-spheres, Calculus of Variations and Partial Differential Equations, to appear. [20] M. Zhu, Uniqueness results through a priori estimates, Part I: A 3dimensional Neuman problem, preprint.
80
A N EXPLICIT E X A M P L E OF STABLE A N D INSTABLE M O T I O N S IN FLUID M E C H A N I C S
Department
Institute
ZHI-MIN C H E N of Mathematics, Tianjin University, Tianjin 300072, P.R. China E-mail: [email protected]
of Applied Mathematics, Beijing 100080,
Chinese Academy P.R. China
of
Sciences
Dedicated to Professor Xiaqi Ding on the Occasion of His 70th Birthday A b s t r a c t . The dynamical behavior of incompressible viscous fluid motions excited by an external sinusoidal force is studied. This forcing gives rise to the occurrence of the basic steady flow wo = (sin 22,0,0). It is shown that there exist exactly twelve global branches of different steady flows bifurcated from uo, which, on the other hand, is local stable in an inifinte-dimensional flow invariant subspace irrespective of the magnitude of the Reynolds number.
1
Introduction
From the well known criteria of Landau [15] and Ruelle and Takens [25], Much of the flow behavior at the verge of transition from laminar to the turbulent state can be understood through a series of steps of bifurcations, yet it is diffi cult even in examining primary bifurcations. For example, the hydrodynamic instabilities of the parallel shear fluid flows have generally been studied by con sidering the Orr-Sommerfeld equations reduced from the linearized equations around a unidirectional basic steady flow (^i(z),0,0)(see Lin [16]). Although many more approaches have been provided (cf. [3,4,11]), there is still lacking exactly analysis for observed primary bifurcations of this basic flow. In 1959, Kolmogorov (see [1]) presented a simpler model describing the two-dimensional spatially periodic fluid motion excited by the sinusoidal ex ternal force k2(sinky,0). This forcing give rise to the unidirectional basic steady flow (sinky,0). The first analysis on this model is due to Mishalkin and Sinai [21] showing the global stability when k = 1 (see also [20] for an alternative approach to this result). Instabilities arise when k > 2 and some critical Reynolds numbers were observed by Iudovich [12](see also [2,18] for the lower bound estimates on Hausdorff dimensions of the associated global attractors), and each of the numbers is connected with a real eigenfunction of the governed linearized Navier-Stokes equations around the basic flow. In fact,
81
the eigenspaces are even-dimensional, and thus more careful examination is necessary for the bifurcation study. To overcome this difficulty, a flow invari ant structure was found for the Kolmogorov's model, which helps to give the existence of steady-state bifurcations and Hopf bifurcations (see [7,8]). For the study of this model from viewpoints of numerical experiment and applied mathematics, one refers to the investigations [22,23] and [10]. Moreover, the well known stability result due to Mishalkin and Sinai [21] has been extended to the three-dimensional case in [17]. One can also refer to Fujita et. al [9] for an interest study on the stability of the two-dimensional Navier-Stokes flows in Annuli. Recently, we obtained some stability and instability results for the threedimensional fluid motion driven by the external force k2(sinky,0,0) for any integer k. In this note, we shall take k — 2 as an special example to describe the nonlinear phenomena we observed recently. More precisely, we consider the three-dimensional spatially periodic fluid motion driven by the external force 4(sin2z,0,0). The hydrodynamic behavior of this fluid motion defined in terms of the velocity u and pressure p is described by the Navier-Stokes equations: dtu - Au + \(u-V)u + AVp = 4(sin2z, 0,0), m where A > 0 represents the Reynolds number defining the viscous fluid motion, and the velocity u satisfies the spatially periodic condition: u(t, x, y, z) = u(t, x + 27r, y, z) — u(t, x, y + 27r, Z) = u(t, x,y,z
+
2TT).
(2)
To ensure the uniqueness of the associated Stokes problem, we require the additional condition: /•27T
/
P2TT
/
/>2TT
/
udxdydz = 0.
(3)
Jo Jo Jo Similar to the shear fluid flows, this problem admits a unidirectional basic steady flow u0 = (sin 2z, 0,0). We intend to specify all of the steady-state solutions bifurcated from I/Q, and on the other hand, to present an infinite-dimensional flow invariant space, in which u 0 is stable irrespective of the magnitude of the Reynolds number. As a by-product of this stability observation, we obtain an example for the global existence of large strong solutions to three-dimensional Navier-Stokes equations. To help formulate the problem, it is convenient to introduce the Hilbert spaces: H2 = {ue L2([0,27r]3;,R3)| Au €
L 2 ([0,2TT] 3 ;
i? 3 ),
82
V • u — 0, u satisfies Eqs. (2-3)} , Hi = lueH2\ + ]C
u=
^(^n,r/n,Cn)sin(2nz)
] C (^n,m,^n,m,Cn,m)C0s(2ny +
2mz)
neN mez + ^2
^^n,m,f}n,rnXn,rn)sm(2ny
+
2mz)
neN mez + 5Z leN
Y2 (€l,n,m,m,n,mXl,n,rn)cOs(2lx n,meZ
+ 2ny +
2mz)
+ 5 Z ] C (6,n,m, ^/,n,m, 6,n,m) sin(2/x + 2ri?/-f 2mz) > , leN n,meZ
J
where TV and Z denote respectively the positive integer set and the integer set. The main results of this paper read as follows: Theorem 1.1 Let A > 0, and let ui e Hi with ||A(^i -
UQ)\\2
< 6,
for some constant 6 > 0. Then Eqs. (1-3) admit a unique solution u in the space C ( [ 0 , O O ) ; # Q )
suc
^
^ai
u(0) = ui and \\A(u(t) - u0)\\2 < ce~3\ where and in what follows \\-\\r denotes the Lr-norm, andc represents a generic constant. Theorem 1.2 There exist two critical Reynolds numbers and twelve branches of steady-state solutions 0 < A0 < Ai and {unA\ A > 0} C H2, n = 1,..., 12 with respect to Eqs. (1-3) such that un,\ = ^o for 0 < A < A o , l < n < 4 , Un,\ 7^ ^m,A ^ U0 for X0 < X,
1 < U / 777, < 4,
un,\ = u0 for 0 < A < Ai, 5 < n < 12, wn,A ^ um,\ ^ u0 for Ai < A, l < n / m < 12, and l|Vtxn>A||2 < ||Vti 0 ||2 =
4TT 3/ 2
for X > 0,1 < n < 12.
(4)
Additionally, there are no other steady-state bifurcation points along the half line {(A,UQ)| A > 0} ; and no other steady-state solutions branching offuo.
83
In fact, there also exist time-dependent periodic solutions branch off the steady state solution u0 (i.e., Hopf bifurcation). For convenience to understand the instability of the fluid motion, we now present in the following such a Hopf bifurcation result, of which the proof is rather lengthy and is to be given elsewhere. Theorem 1.3 Eqs. 1-3 admit two critical Reynolds numbers XH1, XH2 > 0 and six different time-dependent periodic solutions *>I,A, V2,\
for XHl
< X < XHl
+ e
and V3, A, • ■ •, ve,\ for
XH2
< X
AHl
A—>AH2
uo y^ vi,\ ^ v2,\ ^ u0b for XHl < X< XHl H- e, 1
vn,\ i Vm,\ ^ u0 for 3 , nez J
Ei,j,k = . nez J
Ei,j,k = lue I
We see that these subspaces are invariant with respect to A — XA, and this operator is defined by its spectral behavior in these subspaces.
85
Le
mma 2.1 For I G N, j G Z and k = 0,l,
with
0 n = sin(Z:r + jy + kz + 2nz) or cos(lx + jy + kz + 2nz) satisfy the spectral problem Au — XAu — pu = 0 with Rep > — 1. Then {£n} and {r)n} are uniquely determined by {Cn}Proof. After an elementary calculation, the spectral problem Au — XAu — pu = 0 can be rewritten in the form: ] > ^ f ( £ n + P)£n + y ( f n - l
~ f n + l ) + A ( C n - l + C n + l ) J (t>n=~Xdxp,
(5)
+ p)Vn+ir(Vn-l
(6)
nez '52[(Pn
~ Vn+l) ) n=-Xdyp,
(?) XI
SUmneZ
( {Pn + P)(n + y ( C n - 1 ~ C n + l ) J n=-Xdzp,
(8)
^ (f£n + jr/ n + (A: + 2n)Cn)0n=O, nez
(9)
where A* = i8n (l,h k) = l2+ j 2 + (2n + A:)2. From Eqs. (5-6) it follows that Yl
( (Pn + P)(!£n+jr)ny-
nez^
+ I T ( i f n - 1 + j ? M - l ~ i f n + 1 ~ i ^ n + l ) ) n Z
J
= - ^2 A/(Cn-l+Cn+l)0n ~ lX8xp - jXdyp. nez Appllying the operator dz to this equation and the operator — ldx — jdy to Eq. (8) respectively, and summing the resultant equations, we have, for n £ Z, {Pn + p)(l£n+jrin)
+ -J(/£n-l
+ j ^ n - 1 ~ *&i+l ~ J>M+l) J ( 2 n + k)
= - A Z ( C n - l + C n + l ) ( 2 n + *) + ( (/3 n + p)Cn + y ( C n - 1 " Cn+l) ) (Z2 + j 2 ) .
let
86
This together with Eq. (9) implies, for n £ Z,
/3n(^n + p ) C n + y ( C n - l - C n + l ) ( / 2 + i 2 )
= j((2n
+ 2 + fc)C„+i - (2n - 2 + *)Cn-i + 2(Cn-i + Cn+i))(2n + *)
- j((2n
+ 2 + A:)2Cn+i - (2n - 2 + A;)2Cn-i + 4Cn-i ~ 4Cn+i)
From Eqs. (5-6), we may also derive the following equations without the pressure term involved: (Pn + P)U
+ y ( f n - l " f n + l ) + A(Cn-l + Cn+l) J j
= ((Ai+P>7n + y ( 7 7 n _ i - 7 7 n + i ) j /, U E Z. Thus Eqs. (5-9) become the couple set of algebraic equations, for n £ Z, 2/3 n (/? n +p) -Cn A 2(/? n +p)
titn-lVn)
+ l{j£,n-l-lr]n-l)
+ J ( / ? n - l ~ 4 ) C n - l = l{Pn+l
" 4)Cn+l,
(10)
~ / ( i f n + l - f y n + l ) = ~ 2 j ( C n - l + Cn+l), (11)
f f n + j * 7 n + ( 2 n + A;)Cn = 0.
(12)
On the other hand, we shall follow a technique from [21] to show the absence of nontrivial solutions to the couple set of equations
2
xt
n + Tn_1 Tn+1 = n ez
"
°'
'
(13)
Here {r n } is subject to the sumability condition J2nez l r ^l 2 < °°We may suppose rn ^ 0 for all n £ Z, since X ^ G Z l r ^l 2 < °° anc * Eq. (13) with r n o = 0 imply either rn = 0 or -, ^ i r n 0 +n+l | ^ 2(/? nQ+n + Rep) 1> I 1 > —
> oo as n -* oo for n 0 > 0,
87 I T~n0— n—1
1 > |—
T~no — n
,1 ^> 2 ( / 3 n o-—— _ n M+ Rep)
» oo as n -> oo for no < 0,
Dividing Eq. (13) by r n gives - ^ 2 -
=
^
= 0, n > 0,
XI
(14)
r±n
and so 0 as n —> oo.
T
±(n-1)
Applying Eq. (14) repeatedly gives T±n
=Fl
T±(n-1) — 2(P±n + p) A/
r
\ n
for n > 0.
1 2Q9±(w+1)+p) XI
1
Since 0 _ n = fin w h e n A: = 0 a n d (3-n = / 3 n - i w h e n A: = 1, we have T_i
Ti
r0
To
1
2(^1 + p) A/
-, when k = 0,
1 2(02 + p) XI
+
J_
and r_i r0
r0 r_i
1 2(/3o + p) A/
yhen A: = 1.
2(/9i+p) XI
1
Observing that 2(A) + p)
[
A/
T-i
Ti
T0
To
= Q
we have ^
+ i
(
A
+
r t XI
'
I
2(/3 2 + p) A/
= 0 . whe„* = 0, 1
(15)
and 2(A)+ P) + TTTQ \ XI 2(^+p) XI
i
— h when k — 1,
(16)
+
2(fo+p) A/
1
This leads to a contradiction, since the real parts of the right hand sides of Eqs. (15-16) are positive. Thus Eq. (13) has no nontrivial solution. Applying this criterion on Eq. (13) and using the Riesz-Schauder theory, we readily seen from Eq. (11) that {j£n — lr]n} is uniquely determined by {£ n } This together with Eq. (12) implies the desired assertion. The proof is complete. From the proof of Lemma 2.1, we deduce readily the following. Corollary 2.1 There holds, for integers j > 0 and k = 0 , 1 , dim{u e Eo,j,k U ^o,j,fc| Au — XAu — pu = 0} = 0. Now we proceed to study on the spectral behavior of the operator A — XA reduced in Eij^ U Eij^ with / > 1. L e m m a 2.2 Let I G N, j G Z, X > 0 and Rep > -I2 - j 2 . Then dim{u e EtJA
U Ettjil \ Au - XAu - pu = 0} = 0 if Imp = 0,
(17)
dim{u e
U Eljji0\ Au - XAu - pu = 0} = 0 if Imp ^ 0,
(18)
and when (l,j)^
EIJJ0
(1,-1),
(1,0), (1,1),
dim{u € Eiiji0 U Eltji0\ Au - XAu - pu = 0} = 0.
(19)
Proof. With the use of Lemma 2.1, we see that it suffices to deduce the desired assertion by examining Eq. (10). Following the derivation of Eqs. (15-16) from Eq. (13), we deduce from Eq. (10) that, for n ^ 0, (^~4K ±n _ (/?±(n-l)—4)C±(n-l)
=F1 AJ(#fc„-4)
20Mn+1)(0±(n+1)+p) A/(/3±(„+i)-4)
(20)
1
89
Deviding Eq. (10) by (/?„ - 4)£„ implies 2/?n(/3n+/>)
(/?„-! - 4 ) C n - l
AZ(/3„-4) Thus Eq. (10) becomes 2 W O + P) +
M(P0-4)
(P„+l - 4)Cn+l
=
(iS»-4)C„
,
(/3n-4)Cn
l 2A(A+p) AZ(A-4)
+
'
. , . . = . , w h e n * = l,
1 2lh(Jh+p) A/(/3 2 -4)
,01, (21)
1 ..
and
T77^
7T + o/a //a
A/(/3 0 -4)
■—^
2/?i(/?i+p) A/(A~4)
=
1
1 2/?2(/?2 + p) A/(/? 2 -4)
°>
wnen
* = 0-
(22)
1 ..
+
Obviously, Eq. (21) implies Eq. (17). In showing Eq. (18), we follow a technique from [17] to argue by contra diction. Suppose that p with Imp ^ 0 satisfies Eq. (22). We see, for n > 0, |ar
H Ai(fem-4) ; i = i" g (^ ±m +p)i I
, 2/J±(m+i)(/3±(TO+i) + />) A/(/? ± ( m + 1 ) - 4)
and thus, from Eq. (15), ar lars(-T77^—rr)l s( 2/?±i(/?± 1 +p) A/(/?0 - 4) " = 'l 6V
A/(/5±i - 4)
|arg
^ A/(/3±i - 4)
2/? ±2 (l±2 + p) AZ(/9±2-4) +
2/3 ±2 (/3 ±2 + P ) A/(/3 ± 2 -4)
1
1
90 which leads to a contradiction, and arrives at Eq. (18). Moreover, we see that (IJ) f£ { ( 1 , - 1 ) , (1,0), (1,1)} gives/3 0 -4 > 0, which implies that the real part for right hand side of Eq. (22) is positive and thus Eq. (19) is valid. As an immediate application of Lemmas 2.1 and 2.2, we have Corollary 2.2 For A > 0 and Rep > — 4, there holds dim{u e HQ I Au - XAu - pu = 0} = 0. Finally, we present a result on the existence of an eigenvalue p(X) increasing across the imaginary line. Lemma 2.3 Let j = —1,0,1. There exists a function pj : (0, oo) -> R such that dpj(X)/dX > 0, lim pj(X) = - 1 — j 2 , A->-0+
lim Pj(X) — oo, A->-oo
dim{u G #i,j,o| Aw — XAu — pu — 0} < 1, and dim{u G -Ei,.7,01 Au — XAu — pu — 0} < 1 2
for Rep > —1—j and X > 0, where the equalities hold if and only if p — Pj{X). Proof. It follows from Lemma 2.2 that it suffices to consider the existence of the real eigenvalues in the half line (—1 — j 2 , oo). By the proofs of Lemmas 2.1 and 2.2, we see that the spectral problem A — A A — p = 0 reduced in Eijto U Eij,o is equivalent to Eq. (22), that is,
A(/9b-4)
+
j2+4n2.Multiplyingthisequationby
2(3^ + p) A(/3x - 4)
Recall that n (3= /3n(l,j,0) = 1 + -(Po-WPoWo + p))-1 yields
+
2(32(p2+p) A(/? 2 -4)
92{p)
1
1
+
J_ ..
-U"
91
where , x WnM0n+p)(0O+p) 9n{p) = —y- A ^~\nS 7\—>
,
.
,, >
w h e n n 1S o d d
, wQ8w + p ) ( 4 - A ) ) , when # n (p) = -2/?— n is even. Po(Pn - 4 ) ( / % + p)
dg2n(p) ^n i ,o
Denoting by G(A, p) the right hand side of Eq. (24), and observing for n E N, d#2n-i(p) ^n
,
— — > 0 and dp we obtain
—^- < 0, when p > - / 3 0 , dp
Hence the observation lim G(A, p) = oo and lim G(A, p) = 0 implies the uniqueness and existence of p = Pj(A) > — /?o = — 1 — j 2 satisfying | = G(A, Pi (A)).
(26)
Note that d(XG(\,p)) dX Eq. (26) gives for p = Pj(X), d(XG(X,p)) dA
=
> 0 for A > 0 and p > -/30-
d(XG(X,p)) dX
8G(X,p) dp dp dX
dG(X,p) dp dp dX'
This shows, by Eq. (25), dpj/dX > 0. Thus , multiplying Eq. (23) by A gives
Po(J3o + PiW) _ A, - 4 201(Jh+PiW) A 2 (/3i-4)
1
+
2/32(/g2 + Pi (A)) A - 4
1
92 Passing the limits A —> 0 + and A —► oo respectively, we obtain immediately lim pj(X) — — 1 — j
2
and
A—►()+
lim p.-(A) = oo. A—>oo
Now it remains to determine the eigenfunction u = S n 6 Z ( £ n , rjn, Cn)0n to the problem A - XAu - pjU — 0. On setting =fl l±r
2(3n(Pn+Pj(\)) A(/? n -4)
|
1 2/9 (w+1) Q3 (n+1) +p i (A)) A(/3 ( n + 1 ) -4)
1
we have, by Eq. (20),
and thus
(Pn ~ 4)C±n , . n 7±n = 7^ T^T forn>0, (Pn-1 ~ 4)C±(n-l) Co
= c,
where c is an arbitrary constant. From Lemma 2.1, we see that all the eigenfunctions with respect to the eigenvalue pj form a one-dimensional subspace contained in EIJJ0 and £ij,o respectively. The proof is complete. 3
Proof of Theorem 1.1
For convenience, we set A\ = —A + A A Note that A\ is an unbounded operator mapping Hfi into itself and the domain D{A\) is dense in H§. From Corollary 2.2, it not difficult to verify the resolvent estimates \P\ 1Mb + M 1 / 2 ||V U || 2 + ||Au|| 2 < c\\Axu + pu\\2 for u e HQ and Rep > —7/2, where c is independent of u and p. This im plies that the operator A\ generates a strongly continuous analytic semigroup e~tAx, t > 0 in Hi and there holds the following estimates: | | e - ^ | | 2 + ^/2||Ve-M^||2 + *||Ae-MH|2
2([0,2ir]3;R)/R,
we see that iJ 2 ([0,27r] 3 ;i?) is also a Banach algebra for the induced multipli cation. For / G TV and j G Z, we introduce the Hilbert spaces: Hftj([0,2?r]3; i?)4he Banach subalgebra of # 2 ([0, 2TT]3; R) generated by the three modes cos2z, cos(/x -h jy) and cos(lx + jy + 2z), Hij([0,
Banach subalgebra of # 2 ([0, 2TT]3; R) generated by the three modes cos 2z, sin(/x + jy) and sin(/x + jy + 2z),
2TT]3; R}the
H2j = {u=
(LJUUJ2,UJ3)\
UULJ2,U3
e J^.([0,27r] 3 ;fl), V • u = 0} ,
96
Let us present several basic results on this vorticity formulation. Lemma 4.1 Let LJ G H2 be a solution to Eq. (27). Then ||Aw|| 2 < c(A4 + 1) for some constant c independent of X and LJ. Proof. Taking the inner product of Eq. (27) with parts, we have
A~XLJ,
and integrating by
IMIa < IMI2 = 4;r3/2, and hence, by Eq. (27),
||Au|| 2 < A d l A - ^ x w||6||Vu;||3 + I M U I V A ^ V x w || 2 ) + 4||wo||2 < c A | | W | | ^ 4 | | A W | | ^ 4 + 4||u;o||2 Xj is close to Xj. From Lemma 4.3 it follows that (Xj,uo) is the unique bifurcation point on the half line {(A,CJ 0 )| A > 0} with respect to the space Hfj. Thus by the global bifurcation result of Rabinowitz [24], the bifurcated solutions continuously exist for all A > Xj and
99 ^I,J,A ^ 0 /
^2,i,A for A >
Aj.
Moreover we see that LJIJ^X and U;2,J,A are always separated by the stable manifold of LJ0 in H2j. This gives u>ij,\ ^ u2,j,\ for all A > A,. The proof is complete. Following the proof of Lemma 5.1, we have the following. Lemma 5.2 Let Xj with j = - 1 , 0 , 1 be given in Lemma 5.1. Then Eq. (28) admits two branches of solutions U ^ A ^ ^ A € Hfj such that ^3,j,A = ^4,j,\ = ^o, when A < A^^n,j,A 7^ ^m,j,A 7^ ^o 5 wAen A > Aj and 3 < m ^ n < 4. With these preparations, we can now proceed to the proof of Theorem 1.2. Proof of Theorem 1.2. By Corollary 2.1, Lemmas 2.1, 2.2, 2.3,4.3, 5.1 and 5.2, we see that Eq. (28) has and only has twelve branches of solutions bifurcated from LU0. TO obtain the desired properties, we recall from Eq. (22) that Xj satisfies the equation Pl 2 A j ;(A)-4) + - ^ 2/3? + A,-(A-4) '
"
;
= 0,
2%
1
A,-032-4)
..
(29)
where (3n — /? n (l, j , 0) = 1 + j 2 + 4n 2 . This equation yields A_i = Ai. To show the monotonicity AQ < Ai, we rewrite Eq.(29) in the form
1
9i U) + A
i
92(j) +
1 1
fls 0') ,
) =
2/3„ fl> + 4n 2
4 - fl,
- u^m ^' Har-' IT'when n 1S even-
We see that # n (0) <
