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Q2(X)
i
'P(x, v')g2(v')dJ.L(v') E L2(f2),
S(t) is defined by 'Ij; E L2(f2) ---->
i
h(v)e- J:u(x-7"v,V)d7"'Ij;(x_tv)x(t < s(x,v))dJ.L(v) E L2(f2)
(where h(v) = gl(v)h(v)) and 0 1 : 'Ij; E L2(f2)
----> Ql (.)JI (.)'Ij;(.) E
L2(f2 x V).
Hence it suffices to prove that Set) (t > 0) is compact (resp. Set) depends continuously on t > 0 in the uniform topology). By approximating h E Ll(V; dJ.L) by continuous functions with compact supports we may suppose that h is continuous with compact support. Finally, by decomposing h into positive and negative parts, we may suppose that h is non-negative. A basic observation is Set) ~ RSoo(t)E where E: L2(f2)
---->
L2(Rn)
is the extension operator (by zero),
is the restriction operator and
where
Q(V)
= inf a(x,v). xEO
68
Topics in Neutron Transport Theory
Moreover, if n is convex and if a(x , v) = a(v) then (4.16)
Set) = RSoo(t)E.
Thus we are led to show the compactness of RSoo(t) when n is bounded and to appeal to a domination argument [2J . We will also show that Soo(t) is continuous in t > 0 for the uniform topology and this will prove the second claim in view of (4.16). Another basic observation is that, for 'l/J E L2(Jr'),
Soo(t)'l/J
J J
h(v)e-t!l.(v)'l/J(x - tV)d/-L(v)
=
'l/J(x - z)dVt(z) = 'l/J * dVt
=
(4.17)
where dVt is the image of the Radon measure h(v)e-t!l.(v)d/-L(v) under the dilation
v
--+
tv.
According to Lemma 3.2, the compactness of RSoo(t) relies on the condition II
Th(SOO(t)'l/J) - Soo(t)'l/JII
L 2(Rn)
--+
0 as h
--+
for any bounded B c L2(Rn), where Th'l/J(X) Fourier Transform, this amounts to (e ih .( -1)S;::W'l/J11 II
--+
0 as h
--+
0 uniformly in'l/J E B
= 'l/J(x + h). By using the
0 uniformly in 'l/J E B .
L 2(R,)
It is easy to see that there exists a constant c
> 0 depending on B such
that (e ih .( -1)s::;t)'l/J112 ::; 4 II
r
le ih .( - 11
V A> O.
uniformly in 'l/J E B .
(4.18)
IS;::W'l/J12 d(+c sup
J1(I>A
Thus it suffices to prove 2 Is::;t)'l/J1 d(
r
--+
J1(I >A
0 as A
I(I 0) for the uniform topology of
Let t > 0 be fixed. Then
IISoo(t)1/! -
II1/! * (dVt -
Soo(t)1/!IIL2(Rn)
< whence
IISoo(t) -
Soo(t) II
sup (ERn
dl't)II£2(Rn)
Idv't() - dl't() I II 1/! II £2 (Rn)
::; (~UXn IdVt() - iz;-(Ol·
On the other hand,
splits as
It is clear that h () -+ 0 as t -+ t uniformly in ( E Rn. Moreover , h(() and h() are arbitrarily small for 1(1 large enough, uniformly in t belonging to a fixed small neighborhood oft, because the Fourier Transform of h(v)e-t~(v)df,LC1!) goes to zero at infinity. Finally, it is clear that h(()h (() -+ 0 as t -+ t uniformly in ( bounded. HeI1ce
70
Topics in Neutron Transport Theory
which ends the proof of the theorem for p = 2. The case 1 < p < 00 is tackled as follows. We first note that R 2 (t) E L(£P(n x V)) depends continuously on K E L( £P (0. x V)) (uniformly in bounded t). Therefore it suffices to give a proof for a smooth collision operator K, say with a kernel of the form iEI
where I is finite, (ti E LOO(n) and Ii, 9i continuous with compact supports. In such a case R2(t) E L(£T(n x V)) for alII :-:; r < 00 and consequently the above L2 results extend to £P spaces (1 < p < 00) by interpolation. 0
Remark 4.5 We point out that (4.15) is satisfied for the Lebesgue measure on R n (Riemann-Lebesgue Lemma) or on spheres (see, for instance, W. Littman [25]) . The convexity of 0. is essential for the time continuity of R2(')' It is possible however to remove this assumption by imposing a stronger hypothesis on the measure dJ.L (see Corollary 4.3) . Before proceeding further, we point out an interesting by-product of the previous proof. It is a measure theory result which is not (to our knowledge) in the current literature although it looks quite reasonable. In one dimension it is contained in the well-known Wiener's theorem ([20J p. 138) (see also [41J p. 32) . It could probably be deduced from the onedimensional result. It is surprising to prove it so indirectly (via Transport theory!)
Corollary 4.2 Let dJ.L be a bounded Radon measure on Rn whose Fourier transform vanishes at infinity. Then the hyperplanes have zero dJ.L-measure. Proof: Let 0. be bounded and convex. We showed, in the proof of the previous theorem, that K I U(t)K2 is compact in L2(n x V) for t > 0 (for any regular collision operators KI and K2)' It follows that
1
00
K I ()..-T)-IK2 =
e->.tKI U(t)K2dt is compact in L2(n x V).
By choosing K2 = (KI)*' the arguments in the proof of Theorem 4.6 show that K I ().. - T)-I is compact in L2(n x V). This implies, according to Remark 3.1 and Remark 4.1, that the hyperplanes have zero dJ.L-measure which ends the proof. 0 We give now an extension of Theorem 4.7
Theorem 4.8 Let 1 < p < 00 and let the collision operator be regular. We assume that the measure dJ.L is such that translated hyperplanes have zero dJ.L-measure and that 0. is bounded. Then the remainder term R3(t) is compact.
Chapter 4. Spectral analysis. A unified theory
71
Proof: We only consider the case p = 2; the general case follows by approximation and interpolation as in the proof of Theorem 4.7. We deal now with R3(t) = [UK]3 * V. It suffices to prove the compactness of [UK]3. Actually, since [UK]3 = [UK] * [UK]2 = U * (K [U K]2), it suffices to show that
K[UK]2= lotKU(S)KU(t-S)KdS is compact. By approximating the regular collision operator K, it suffices to consider
lot K 1 U(s)K2U(t - S)K3ds where Ki (i = 1, 2,3) have kernels
In such a case,
where 02 E
L(L2(n x V), L2(n))
S2(t) E L(L2(n), L2(n)) Sl(t) E L(L2(n), L2(n))
01
E
L(L2(n), L2(n x V))
act , respectively, as follows
cp
-t
f
[
D:2(X) V h2(V) e
f
[
cp-t . vh1(v) e
cp
-t
_It
D:l(x)cp(x)h(v)
0
_It
U(X-Tv,v)dT
0
U(X-Tv,v)dT
1
cp(x - tv)x(t < s(x, v))
1
cp(x-tv)x(t < s(x,v)) dp,(v)
72
Topics in Neutron Transport Theory
and h2 = g2/J, hI = glh By approximating hi (i = 1,2) by continuous functions with compact supports (and decomposing them into positive and negative parts) we may suppose that hi (i = 1,2) are continuous functions with compact supports and non-negative. Thus we are led to prove the compactness of lot SI(s)S2(t - s)ds.
It is not difficult to see that
where
c = IIa31100 IIhI li oo IIh21100 and e' SUpphi C
is a constant such that
{v; Ivl::; e'}
(i = 1,2) .
We denote by Rand E, respectively, the restriction operator to n and the extension operator to Rn. Thus, by domination [2] , it suffices to prove the compactness of Riot S(s)S(t - s)ds.
We introduce the measure df3(v) S(t)cp
=
J
(4.19)
= 1{l v l:5 c ,}d/-l(v).
cp(x - tV)df3(v)
=
Then
J
cp(x - z)df3t(z)
where df3t is the image of df3 under the dilation v (4.19) is nothing but
--t
= df3t * cp
tv. Thus the operator
where da
= lot df3s * df3t-sds.
(4.20)
Arguing as in the proof of Theorem 4.7, it suffices to show that
(4.21)
73
Chapter 4. Spectral analysis. A unified theory where
z is the Fourier Transform of the measure da. We note that
z ( < )=
bf
[I
[J
~ ( ( ) d ~ ~ (=~ o) d se - h . c d ~ s ( ~ ) ] e-'~.~d/3~-,(y)]ds,
Thus
z(O
=
/J
e-fi.C - e-"z.C i("
-
y1.C
Introducing the polar coordinates C = ICI w , w as
dP(x)dP(~). E
(4.22)
Sn-l, we decompose (4.22)
Hence
where
Thus, to prove (4.21) , it suflices that dP(x)dp(y) = 0 uniformly in w E Sn-'. We note that
so, by the dominated convergence theorem, it suffices to show that for each y E Rn SUP
wESn-'
1
l(z-y).wll€
dp(x) -,o as E
-,0.
74
Topics in Neutron k s p o r t Theory
Using the compactness of 9 - l ( a s in Lemma 3.1), this amounts to
J
dp(x) -+ 0 as E -+ 0 for each w E sn-land y E Rn
I(x-Y).~~E
and this means that dp {x; (x - y).w = 0) = 0,i.e. translated hyperplanes have zero dp-measure. Findy, this is equivalent to saying that translated hyperplanes have zero dp-measure. 0 The previous theorems provide us with sufficient conditions for the compactness of R2(t) or R3(t). Actually, it is possible to give a general condition under which &(t) (m 2) is compact. However, this condition is of abstract character. This is the reason why we state this general result separately. To this end we introduce some notations. For each c > 0 we d e h e a truncation of the measure dp
>
Let M(Rn) be the space of bounded Radon measures on Rn. For each couple t ~ ( 0 , o o ) - - + d a i ( t ) € M ( R n )i = 1 , 2 of measure-valued and strongly continuous mappings, we define their convolution as t dal * do? = dal(s) * da2(t - s)ds
1
where the sign * under the integral sign is the usual convolution of measures. We can also define the repeated convolution of t E (0, oo) + da(t) E M(Rn) and set
[da(t)lm= da * . . - * d a (m times).
Then we have
.- Theorem 4.9 Let 1 < p < oo and let the collision operator be regular. We assume that R is bounded. For each 0 < c < oo we define dp(v) by (4.23). Let dBt be the image of dp under the dilation v -+ tv (t > 0). If there exists an integer m 2 2 such that the Fourier Transform of [dptlm-I vanishes at infinity, then &(t) is compact.
Proof: As previously, we only consider the case p = 2. Since &(t) [UKIm* V , it suffices to show the compactness of
=
75
Chapter 4. Spectral analysis. A unified theory
Again, it suffices to prove the compactness of K [U K]m-l . On the other hand, by the usual procedure, it suffices to consider (4.24) where ki(x,v,v') = ai(x)fi(v)gi(v')
(1 ~ i ~ m)
with ai E Loo(n); Ii,gi continuous with compact supports and nonnegative. Let c > 0 be such that the supports of Ii, gi (1 ~ i ~ m) are included in {v; Ivl ~ c } and let m-l
'Y
Then
=
m-l
m-l
(II lIaill II Ilhll II Ilgill i=2 i=2 i=2 oo )(
Kl [U K 2] * ...
oo )(
* [U Km]
oo )'
~ 01R [d,Bt]m-l E02
where [d,Bt]m-l denotes symbolically the operator of convolution with the measure [d,Bt]m-l, O 2 : 'ljJ E L2(n x V)
-+
am (x)
J
'ljJ(x, V')gm(v')dp,(v') E L2(n)
and 0 1 : t.p E L2(n)
-+
al(x)fI(v)t.p(x) E L2(n x V).
Finally it suffices to prove that R [d,Btl m -
1
:
L2(Rn)
-+
L2(Rn)
is compact
and this is a consequence of the assumption on the measure [d,Bt]m-l . We close this section with a converse theorem in the spirit of Theorem 4.5. We note that if, for some integer m, Rm(t) is compact in V(n x V; dxdp,(v)) for all t > 0 then (see Theorem 2.8) [(>. - T)-1 K]m+1 is compact in V(n x V; dxdp,(v)) and consequently, by Theorem 4.5, Km+l is compact in V(V; dp,) . Actually the following stronger result holds Theorem 4.10 Let 1 < p < 00 and let K E L(V(V)) be non-negative. We assume that V is bounded and that 0. is convex and bounded. Let there exist to > 0 and m ~ 1 such that Rm(t) is compact in V(n x V;dxdp,(v)) for t E [0, to] . Then K m is compact in V(V; dp,).
Before giving a proof, we need a preliminary result. For each Sen, we define the multiplication operator Qs : 'ljJ(
)
-+
ls(x)'ljJ(x, v) .
76
Topics in Neutron nansport Theory
Lemma 4.2 Let S1 and S2 be two open subsets of R such that Then, for all $ E LP+( V ;d p ) ,
QslU(s)Qs2$ > e-sllullm &sl$ i f s l
as2)
dist ( S 1 ;
vo
C
Sz.
(4.25)
where vo is the maximum speed. Proof: Let $ E LP+(V;d p ) and let cp = Qs,$J. Then
It is clear that if x t S1 and if s 5
d i s t ( ~ a s 2 )then
x - sv E S2. Hence
which ends the proof. 0 Proof of Theorem 4.10 : Let {So,...,Sm+l) be m+2 convex open subsets of R such that for (i = 0, ...,m). Let
c SSil
Let T 5 min(to, c). By assumption, &@) is compact in L P ( R x V ;d x d p ( v ) ) . Moreover
k@) = [UKIm*V > [UKIm*U
because Qsi 5 I and V 2 U . According to Lemma 4.2,
QS,U(S)QS,+~~L~(1 V )e-sl'ullm Qs, . Hence
77
Chapter 4. Spectral analysis. A unified theory Similarly, the last term of (4.26) dominates
Qs",_, [e- sllulI "",
r
K.
By induction, one sees that
Rm(I)ILP(V) ~ Qso [e-sllull"",]m (I)Km
* ... * (e-sllull"",)
where [e- sllull "",] m = (e-sllull"", )
PSo : 'Ij; E £P(n x V)
---->
(m times) . Let
r 'Ij;(x, v)dx
iso
E £p(V) .
Then the compact operator PsoRm(I)ILP(V) dominates the following operator meas(So) [e- sllulI "", m (I)Km
1
and consequently K m is compact in £p(V; df-L) by domination [2] . (;
4.4
Evolution problems in L1
We begin with a continuity result which does not rely on the convexity of
n. Proposition 4.2 Let K be a regular collision operator on Ll(n x V). Let
a(x, v) For each 0
= a(v)
and t
-t
a(tv) be continuous for each v -j. O.
< c < 00 we set d{3(v) t
= l{lvl~c}df-L(v) and assume that
> 0 - t d{3t
is continuous in the uniform topology (i. e. the norm of total variation) where d{3t is the image of d{3 under the dilation v - t tv. Then R 2 (t) is continuous in t ~ 0 for the uniform topology.
Before proving this result, let us show that the Radon measures satisfying the assumptions above "are" those which are absolutely continuous in "speeds" but arbitrary in "angles". This excludes the usual multigroup models. Proposition 4.3 Let df-L(v) = da(p) ® d{3(w) where v = pw, w E sn-l, d{3 is a Radon measure on Sn-l and da(p) = h(p)dp (h E £1(0, (0)). Then t > 0 - t dp,t is continuous in the total variation norm.
78
Topics in Neutron Transport Theory
Proof We note that IId/ltll :::; Iid/lil :::; IIhll£l(O.oo) Ild,Bll. Thus d/lt depends (linearly and) continuously on h E Ll(O,oo), uniformly on t > O. Then, by approximation, we may suppose that h is continuous with compact support. We fix t > 0 and take continuous test functions cp with compact supports, (d/lt,cp)
=
J
cp(v)d/lt(v)
=
J
(d/l t , cp)
=
IsSn-l
Thus
and
(d/lt - d/lr , cp) =
1 hn-l 00
cp(tv)d/l(v)
=
1
00
d,B(w)
JJ
cp( TW)
0
cp(tpw)h(p)dpd,B(w).
T dT cp(Tw)h( - ) t t
[rl h(~) - rl h(1)] dTd,B(w) .
Therefore
Iid/lt-d/lrll:::; roolrlh(!.)_rlh(~)ldT.
Jo
t
t
r
Jsn-l
d,B(w)---->O ast---->t
in view of the smoothness of h. Remark 4.6 The assumption in Proposition 4.2 is not satisfied by the Lebesgue measure on spheres. Indeed, let (for instance) d/l be the Lebesgue measure on sn-I and let 0 < t < t. We choose a continuous function cp with compact support such that Icp(z)1 :::; 1, cp(z) = 1 if Izl = t and cp(z) = -1 if Izl = t. Then
and d/lr(cp)
so
that Iid/l t
-
= hn-l cp("tw)d/l(w) = -Isn-Il
d/lrll = 2lsn-11 Vt < t
Proof of Proposition 4.2 : We know that R 2(t) = [U K]2 thanks to Lemma 2.3, to prove that t > 0 ----> KU(t)K
* V. It suffices,
is continuous in the uniform topology.
By approximating K by separable kernels, we are led to study K I U{t)K2 where ki(x, v, Vi) = O!i{X)fi(V)9i{V') (i = 1,2), O!i E LOO{n), Ii E LI{V), gi E Loo(V) . We factorize K I U(t)K2 as 0IS(t)02 where
02: 'I/J E LI(n x V) ----> 0!2{X)
Iv
'I/J(x, V' )g2(V')d/l{v') E LI{n),
79
Chapter 4. Spectral analysis. A unified theory Set) is the operator cp E L1(n)
--+
Iv
e-tCT{V')cp(x - tv')h(v')l{t<s{x.v,)}dl.£(v') E L1(n)
(h = g112) and 0 1 : cp E L1(n)
--+ Ct1 (x)cp(x)h (v) E
L1(n x V) .
Finally, it suffices to study Set) and to assume that h is continuous with compact support. Let t > 0 be fixed. We note that
S(t)cp
J =J =
e-tCT{v')cp(x - tv')h(v')l{t<s{x,v,)}dl.£(v') e-tCT{v')cp(x - tv')h(v')1{1<s{x.tv,)}dl.£(v')
so that
S(t)cp
J =J =
cp(x - z)1{1<s{x.z)}e- tCT {f) h( I )dl.£t(z)
cp(x - z)1{1<s{x.z)}q(t, z)dl.£t(z)
where dl.£t is the image of dl.£ under the dilation z e-tCT{f} h( I) ' Hence
II S(t) - S(1)11 L{L'{!1) ::; Ilq(t, z)dl.£t(z) -
--+
tz and q(t, z)
q(t, z)dI.£I(z)
II
(4.27)
where the right-hand side of (4.27) (a total variation norm) is estimated by
Ilqllv'" IIdl.£t - dl.£III
+
J
Iq(t, z) - q(t, z)1 dl.£t(z)
(4.28)
where the first term of (4.28) goes to zero as t --+ t by assumption while the second one goes to zero by the dominated convergence theorem. (v) then R2(t) is compact in £1(0 x V).
Proof Since R 2(t) = [U K]2 * V and [U K]2 = U * (KU K) it suffices, thanks to Theorem 2.2 and Theorem 2.3, to consider KU(t)K (t > 0). By the usual approximation procedure, we may restrict ourselves to K 1 U(t)K2 where
By factorizing K 1 U(t)K2, as in the proof of Proposition 4.2, it suffices to consider the operator Set)
where (h = gl12). By decomposing h into positive and negative parts, we may assume that h is non-negative. Let
0
where IAI is the Lebesgue measure of the set A . Then R3(t) is weakly compact in Ll(D x V) . Proof Let K be a regular collision operator. We observe that R3(t) = [UKj3 * V and [UKj3 = U * (K [UKj2) so, according to Theorem 2.3 (or Theorem 2.4), it suffices to prove the weak compactness of
lot KU(s)KU(t - s)Kds . By approximating K by separable kernels, we may restrict ourselves to
lot K U(s)K2U(t - s)K3ds 1
(4.32)
where ki(x, v, v') = Cti(X)Ii(V)gi(V'), Cti E LOO(D), Ii E Ll(V), gi E Loo(V) (i = 1,2, 3) . One sees that (4.32) is factorizable as
where
and Si(S) E L(Ll(D)) (i = 2,1) act, respectively, as
r.p
-+ Ct2
r.p while
-+
( )ivrr.p( )-is -i s ivr X
x - sv e
r.p(x - sv)e
a
a
u(x-rv,v)dr
u(x-rv,v)dr
( ) 1(s<s(x,v»h 2 v dJl(v)
l(s<s(x,v»hl(V)dJl(v)
84
Topics in Neutron Transport Theory
(hI = gIi2 ,h2 = g2!3 E LI(V)) . It suffices to consider lot SI(S)S2(t - s)ds.
(4.33)
By approximation (and decomposition) we may assume that hi (i = 1,2) are continuous with compact supports and non-negative. One verifies that the operator (4.33) is dominated by the operator
r.p E LI(fl)
--+
R (lot d/it * d/it-sdS)
* Er.p
where Rand E are the usual restriction (to fl) and extension (to Rn) operators,
d/i = l{lvl:5c}dp,(v), the constant c being such that supp(h i ) C {Ivl ::; c} (i = 1,2), * denotes the convolution of measures and d/it is the image of d/i under the dilation v --+ tv. It suffices to show the compactness of
r.p E LI(Rn)
--+
R (lot d/it * d/it- sdS)
* r.p E LI (Rn) .
To this end it suffices that J~ d/it * d/it-sds be a (U) function (see, for instance, [6] p. 74). By the Radon-Nikodym theorem ([39] ,p. 117), this amounts to
i.e.
which means that
i.e.
r i1xl.lyl :5C
[ rt 1A(sx
io
+ (t -
S)Y)dS] dp,(x)dp,(y)
--+
0 as
IAI
--+
O.
(4.34)
By the fact that dp, {O} = 0, one sees that (4.34) is equivalent to the assumption (4.31). This ends the proof when K is regular. If K is nonnegative and (only) dominated by a regular collision operator, the result follows from the previous one by a domination argument. 0
85
Chapter 4. Spectral analysis. A unified theory
4.5
The effects of delayed neutrons
The transport equations considered in the previous sections describe scattering phenomena and also the production of additional neutrons by fissions provided they appear instantaneously (prompt neutrons). However , in fissile materials some neutrons may appear after a time delay as the decay products of radioactive fission fragments. In such a case, the appropriate equation of the neutronic distribution is the following 8fo 8fo at + v . 8x + a(x, v)fo
J
k o(x , v,v' )fo(x , v' , t)dJ1-(v')
+ fAdi
(4.35)
i =l
subject to boundary and initial conditions fo( .,. , t)IL = 0, fo(x, v , 0) = fO(x, v)
where the distributions fi (1 ~ i ~ m) of the delayed neutron emitters (precursors) are governed by the differential equations dfi dt fi(x , v , 0)
-Adi
+
J
ki (x , v , v')fo(x,v',t)dJ1-(v')
fi(x , v) ; (1 ~ i ~ m)
(4.36)
and the positive numbers Ai are the radioactive decay constants [26J . It is possible to solve explicitly fi (1 ~ i ~ m) in terms of fo by means of (4.36) and to insert their expressions in (4.35) . This enables us to deal with only one equation. However, such an equation containing a memory term is not well-suited to the spectral theory. Thus it is more convenient to deal with the coupled system (4.35) (4.36) . To show its well-posedness we write it in the vector form
dw
dt = Aw
; w(O)
= Wo
(4.37)
where W = (Jo , iI , ... , fm)J. and A is the (m+ I, m+ I)-matrix of operators
A=
Topics in Neutron Transport Theory
86
:s :s
where To is the streaming operator defined in section 4.1, Kj (0 j m) represent the integral operators with kernels k j (x, v, v') and I is the identity operator. The functional setting is [£P(r! x V)Jm+l ; 1
:s p
0) in the proof of Theorem 4.7. We consider the strong integral
We recall that, according to Corollary 4.2, the hyperplanes have zero dll'measure and consequently Ki(A - TO)-1 are compact in view of Theorem 4.1. This may be expressed as Ki : D(To)
-7
LP(f! x V) are compact (1 ::; i ::; m)
where D(To) is equipped with the graph norm. On the other hand, by a general result from semigroup theory,
Hence
r f: Aie-A;(t-s) KiUo(s)ds
Jo
is compact.
i=1
Thus i t BU(t - s)BU(s)Bds is a compact operator and consequently [U B]3 is also compact. ·T his implies the compactness of R3(t) from Theorem 2.6 and the stability of the essential type from Theorem 2.10. Remark 4.9 Note that the unperturbed semigroup is uncoupled so its essential type is equal to max{1],-Al, .. . ,-Am } because e-A;t (1::; i::; m) are eigenvalues of U(t) of infinite multiplicities. The assumption on the measure dJ.-L covers the continuous and the multigroup models. Remark 4.10 The spectral theory in £1 setting carries over the same lines using the Ll compactness results of the previous sections.
4.6
Comments
The material in this chapter is an expanded version of M. Mokhtar-Kharroubi [35]. Note that Theorem 4.1 was proved by V.S. Vladimirov [48] for a
91
Chapter 4. Spectral analysis. A unified theory
monokinetic model by exploiting the dissipativity of the streaming operator. His argument was later used, for general continuous models by M. Borysiewicz and J. Mika [5], M. Mokhtar-Kharroubi [30] [33] and (in the context of linearized Boltzmann equation) by A. Palczewski [36]. Note that this argument is also adaptable to time-dependent problems and nonhomogeneous boundary conditions (see K. Jarmouni-Idrissi and M. MokhtarKharroubi [16]) . Compactness results for stationary transport operators, with a view to spectral problems, are also given, for instance, by S. Ukai: [44], S. Albertoni and B. Montagnini [1], J . Mika [27], E.W. Larsen and P.F. Zweifel [21]. Compactness results for multigroup models are given by K. Jorgens [17], G.H. Pimbley [37], M. Mokhtar-Kharroubi [32] . As regard to compactness for the spectral analysis of transport semigroups, besides the fundamental work of K. Jorgens [17] (when the velocity space is bounded away from zero), the basic ideas are due to I. Vidav [47] who introduced, among other ideas, continuity conditions (for the uniform topology) for operators of the form U(t1)KU(t2)K. ..u(tm)K ( ti > ,1 < i < m ) in order to recover compactness results for remainder terms of the DysonPhillips expansion (the strong convex compactness theorem (Theorem 2.2) was unknown at that time). Such continuity assumptions were somewhat neglected (except in Y. Shizuta's paper [40]) and considered difficult to satisfy for usual scattering kernels. In fact , such continuity assumptions turned out to hold for continuous models and regular collision operators (M. Mokhtar-Kharroubi [34]) . Actually they hold for much more general measures (see the proofs of Theorem 4.7 and Proposition 4.2). Compactness results for remainder terms of the Dyson-Phillips expansion are given by J . Voigt [50] who factored the second-order remainder term in order to avoid Vidav's continuity assumptions. Furthermore, J. Voigt [50] deals with possibly unbounded domains. This factorization technique was also used by M. Mokhtar-Kharroubi [33]. We point out that factoring techniques are also fully exploited by M. Borysiewicz and J . Mika [5] and Y. Shizuta [40] . Domination arguments, to prove the weak compactness in L1 of the second-order remainder term of the Dyson-Phillips expansion, were used by G. Greiner [12], J. Voigt [50] and P. Takak [43]. The use of domination arguments was systematized by M. Mokhtar-Kharroubi [33] and exploited to derive inverse compactness results. Finally, we point out the paper by 1. Weis [52] where the strong convex compactness theorem (Theorem 2.2) is given to deal with the compactness of the second-order remainder term of the Dyson-Phillips expansion thus avoiding Vidav's continuity assumptions. The spectral theory of transport generators with delayed neutrons was studied only for simple models in slab geometry by J. Mika [28] and H.G. Kaper [19] while, to our knowledge, the spectral theory of transport semigroups with delayed neutrons has never been studied. The material
°
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in Section 4.5 is an improved version of [31] Chapter VI. One of the main concerns of this chapter was an attempt to unify and extend the known compactness results. As the different direct and inverse compactness results of this chapter show, the spectral theory of T + K or et(T+K) is intimately related to the compactness of the collision operator K with respect to velocities. At this point we mention interesting open problems of physical and (or) mathematical interest. Before stating them we recall that, in solid moderators, the collision operator K is a sum of (an incoherent part) Ki and (a coherent part) Kc where Ki is compact in L2(V) while K c is not [5] . We first state an easy extension of a result by M. Borysiewicz and J . Mika [5]. Theorem 4.16 Let 1 < p < 00 and let dJ1. be a Radon measure such that the hyperplanes have zero dJ1.-measure. Let K = K1 + K2 be a collision operator such that K2 is regular. Let ,8 = inf{ a ;a>
7],
ru((A - T)-l K 1)
< 1 VA> a} .
Then (J(T + K) n {A; ReA>,8} consists of, at most, isolated eigenvalues with .finite algebraic multiplicities. Proof: Let ReA >,8. We consider the problem At.p - Tt.p - K t.p = S ,
i.e. which is equivalent to t.p - (A - T)-l K1 t.p - (). - T)-l K 2t.p = (A - T)-l S.
(4.42)
Since ru((A -T)-lK1) < 1, (4.42) is equivalent to t.p- [I
- ().. -
1 T)-1 K1r ().._T)-1 K2t.p =
[I - ().. -
1
T)-l K1r ()..-T)-lS.
From Theorem 4.1, ()..-T)-lK 2 is compact in V'(D. x V ;dxdJ1.(v)). Hence L()..) =
[I - ().. -
1
T)-l K1r ().. - T)-lK2
is a holomorphic family of compact operators. It follows from GohbergShmulyan's theorem [38] that 1- L()') is invertible except for a discrete set 3 C {A; Re).. > ,8} consisting of degenerate poles of (I - L()..))-l . Thus, for Re).. > ,8, ).. 1. 3, ().. - T - K)-l = (I - L()..))-l [I - ().. - T)-l K 1] -1
().. _
T)-l
Chapter 4. Spectral analysis. A unified theory
93
and :=: consists of isolated eigenvalues of T + K with finite multiplicities. We start with three problems in connection with Theorem 4.16: Problem 1: What is the structure of the spectrum of T + K in the region {TJ < ReA < .B} ? It is clear that we cannot expect interesting results in a such abstract setting. It is more useful to try concrete non-compact collision operators in the spirit of those considered by E .W. Larsen and P.F. Zweifel [21]. We point out that in £1 spaces, K 2 ()..-T)-1 is not weakly compact [13] . The weak compactness of ().. - T)-l K2 is open. Therefore the following problem seems also to be open. Problem 2: Is Theorem 4.16 true in L1 spaces? The following problem seems to be open even in LP spaces. Problem 3: Under the assumptions of Theorem 4.16, find (an estimate of) the essential type of et(T+K) . This is, of course, necessary in order to understand the time asymptotic behaviour of et(T+K) . In section 4.3, several compactness results about the remainder terms J4n(t) are given. The first one (Theorem 4.7) is based on the assumption that the Fourier transform of (the truncations of) the Radon measure dtL goes to zero at infinity. The second one (Theorem 4.8) relies on the assumption that translated hyperplanes have zero dtL-measure. Finally, the third one (Theorem 4.9), which i3 the most general, is based on the abstract assumption on dtL ensuring that the Fourier 'Transform of [d.Bt]m (for some integer m :::: 1) vanishes at infinity. Hence the natural question: Problem 4: Is it possible to translate in simple geometrical terms, as in Theorem 4.8, the condition in Theorem 4.9 ? The reader has certainly observed that the compactness results rely on different assumptions on the measure dtL depending on whether we deal with the spectral analysis of the generator T +K or the semigroup et(T+K) . Note that, for the usual Lebesgue measures on Rn or on spheres, both assumptions are satisfied. It seems that this is the reason why no counterexample to the (partial) spectral mapping theorem (4.43) has been found till now. We recall that the lack (in general) of such spectral mapping theorems was the initial motivation (see I. Vidav [47]) for studying the spectrum of the transport semigroup instead of that of its generator and was a recurrent theme in the subsequent literature on this topic. The results of this chapter seem to indicate that the first step in the direction of a possible counterexample to (4.43) should be to solve the following tricky question:
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Topics in Neutron Transport Theory
Problem 5: Find a Radon measure dp, such that the hyperplanes have zero dp,-measure and such that, for all m ~ 1, the Fourier Transform of [d.Bt]m does not vanish at infinity. We also see that the assumptions on the measure dp, depend on whether we consider V spaces (1 < p < 00) or L1 spaces, whence the natural question: Problem 6: Is (). - T)-1 K power compact in L1 under the assumption that dp, is such that the hyperplanes have zero dp,-measure? Is it possible that, for certain measures dp" the structure of a(T+K)n(Re). > ry) depends on whether p > 1 or p = 1 ? Finally, we mention a plausible conjecture: Problem 7: Prove that the .first remainder term R 1 (t) = V(t) - U(t) is not compact (or weakly compact in L1) in general. We note that, even if the continuous and multigroup models display the same asymptotic spectral structure, they enjoy different compactness properties. Indeed, R2(t) is compact for the continuous model but probably not compact for the multigroup one (see Remark 4.7 and Proposition 4.4). This indicates that different measures dp, should induce different smoothing effects. We note also that the presence of delayed neutrons has an effect on the compactness properties since R3(t) is compact (Theorem 4.15) while R2(t) is (probably) not compact. It is to be noted that the spectral theory in unbounded domains (e.g. half-spaces or the whole space) is much less studied. However, for compactly supported cross-sections many connections exist with spectral theory in bounded domains (see A. Huber [15]). We also note that the spectral mapping theorem (4.43) holds (in an abstract setting) if t > 0 --? KU(t)K is continuous in the uniform topology (see F. Andreu, J . Martinez and J .M. Mazon [3]) and this assumption is actually satisfied by transport operators, regardless of the boundedness of n, under very general assumptions as is shown in the proofs of Theorem 4.7 and Proposition 4.2. The spectral mapping theorem (4.43) probably holds under even much more general conditions (see Conjecture 2.1) . We mention, though it is not the purpose of this book, the existence of an important spectral literature in connection with the Boltzmann equation particularly in the Japanese School (see, for instance, Y. Shizuta [40] , S. Ukai: [45], A. Palczewski [36], C. Cercignani [7] , N. Bellomo, A. Palczewski and G. Toscani [4] and references therein) . There is also a general spectral theory for one-dimensional transport operators with abstract boundary operators by K. Latrach [22] [23] [24]. In this chapter, we dealt only with the general aspects of spectral theory which are consequences of compactness results. The peripheral spectral theory, in connection with positivity, is considered in Chapter 5. Positivity also played a decisive role in the present chapter, in the analysis of compactness via domination theorems [2] [8], even if no
Chapter 4. Spectral analysis. A unified theory
95
positivity assumption is made on the collision operators. This is due to the fact that regular collision operators can be approximated by differences of positive collision operators. We will also consider much finer aspects of the point spectrum in Chapter 6 for form positive collision operators.
References [1] S. Albertoni and B. Montagnini. On the spectrum of neutron transport equation in finite bodies. J. Math. Anal. Appl. 13 (1966) 19-48. [2] D. Aliprantis and O. Burkinshaw. Positive compact operators on Banach lattices. Math. Z. 174 (1980) 289-298. [3] F. Andreu, J. Martinez and J.M. Mazon. A spectral mapping theorem for perturbed strongly continuous semigroups. Math. Ann. 291 (1991) 453-462. [4] N. Bellomo, A. Palczewski and G. Toscani. Mathematical Topics in Nonlinear Kinetic Theory. World Scientific Publishing, 1988. [5] M. Borysiewicz and J. Mika. Time behaviour of thermal neutrons in moderating media. J. Math. Anal. Appl. 26 (1969) 461-478. [6] H. Brezis. Analyse Fonctionnelle:Theorie et Applications. Masson, Paris, 1983. [7] C. Cercignani. The Boltzmann equation and its applications. Springer Verlag. Appl. Math. Sci. 67, 1988. [8] P. Doods and D.H. Fremlin. Compact operators in Banach lattices. Israel J. Math. 34 (1979) 287-320. [9] J.J. Duderstadt and W.R. Martin. Transport Theory. John Wiley & Sons, Inc, 1979. [10] N. Dunford and J.T. Schwartz. Linear Operators, Part 1. Interscience, 1958. [11] W. Greenberg, C. Van der Mee and V. Protopopescu. Boundary Value Problems in Abstract Kinetic Theory. Birkhiiuser Verlag, 1987. [12] G. Greiner. Spectral properties and asymptotic behavior of the linear transport equation. Math. Z. 185 (1984) 167-177.
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[13] F. Golse, P.L. Lions, B. Perthame and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. [14] B. Host, J.F. Mela and F. Parreau. Analyse harmonique des mesures. Asterisque, 135-136. Societe MatMmatique de France, 1986. [15] A. Huber. Spectral properties of the linear multiple scattering operator in LI-Banach lattices. Int Eq. Op Theory 6 (1983) 357-37l. [16] K. Jarmouni-Idrissi and M. Mokhtar-Kharroubi. Dissipativity of transport operators and regularity of velocity averages. Work in preparation. [17] K. Jorgens. An asymptotic expansion in the theory of neutron transport. Comm. Pure. Appl. Math. 11 (1958) 219-242. [18] H.G. Kaper, C.G. Lekkerkerker and J. Hejtmanek. Spectral Methods in Linear Transport Theory. Birkhiiuser Verlag, 1982. [19] H.G. Kaper. The initial-value transport problem for monoenergetic neutrons in an infinite slab with delayed neutron production. J. Math. Anal. Appl. 19 (1967) 207-230. [20] Y. Katznelson. An introduction to Harmonic Analysis. Dover Publication, Inc, 1976. [21] E.W . Larsen and P.F. Zweifel. On the spectrum of the linear transport operator. J. Math. Phys. 15 NOll (1974) 1987-1997. [22] K. Latrach. Theorie spectrale d'equations cinetiques. These, Universite de Franche-Comte Besanc;on, 1992. [23] K. Latrach. Compactness properties for linear transport operators with abstract boundary conditions in slab geometry. Transp . Theory. Stat. Phys. 22 (1993) 409-430. [24] K. Latrach. Quelques remarques sur Ie spectre essentiel et application a l'equation de transport. C.R. Acad. Sci. Paris. Ser J. 323 (1996) 469-474. [25] W . Littman. Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Amer. Math. Soc. 69 (1963) 766-770. [26] J .T. Marti. Mathematical foundations of kinetics in neutron transport theory. Nucleonik. 8, Bd, Helft3 (1966) 159-163.
Chapter 4. Spectral analysis. A unified theory
97
[27] J. Mika. Time dependent neutron transport in plane geometry. Nucleonik. 9, Bd, Helft4 (1967) 200-205. [28] J . Mika. The effects of delayed neutrons on the spectrum of the transport operator. Nucleonik. 9 Bd, Helft1 (1967) 46-5l. [29] B. Montagnini and M.L. Demeru. Complete continuity of the free gas scattering operator in neutron thermalization theory. J. Math. Anal. Appl. 12 (1965) 49-57. [30] M. Mokhtar-Kharroubi. La compacite dans la tMorie du transport des neutrons. C.R. Acad. Sci. Paris. Ser I. 303 (1986) 617-619. [31] M. Mokhtar-Kharroubi. Les equations de la neutronique. These d'Etat, Paris, 1987. [32] M. Mokhtar-Kharroubi. Spectral theory of the multigroup transport operator. Eur. J. Mech. B/Fluids. 9 N0 2 (1990) 197-222. [33] M. Mokhtar-Kharroubi. Time asymptotic behaviour and compactness in transport theory. Eur. J. Mech. B/Fluids. 11 n 0 1 (1992) 39-68. [34] M. Mokhtar-Kharroubi. Effets regularisants en theorie neutronique. C.R. Acad. Sci. Paris. Ser I. 309 (1990) 545-548. [35] M. Mokhtar-Kharroubi. A unified treatment of the compactness in neutron transport theory with applications to spectral theory. Publications mathematiques de Besan{:on, 1995-1996. [36] A. Palczewski. Spectral properties of the space inhomogeneous linearized Boltzmann operator. Transp. Theory Stat. Phys. 13 (1984) 409-430. [37] G.H. Pimbley. Solution of an initial value problem for the multivelocity neutron transport equation with a slab geometry. J. Math. Mech. 8 (1958) . [38] M. llibaric and I. Vidav. Analytic properties of the inverse A(z)-l of an analytic linear operator valued function A(z) . Arch. Rational Mech. Anal. 32 (1969) 298-310. [39] W. Rudin. Analyse reelle et complexe. Masson, Paris, 1978. [40] Y. Shizuta. On the classical solutions of the Boltzmann equation. Comm. Pure. Appl. Math. 36 (1983) 705-754.
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[41] E.M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971. [42] S. Steinberg. Meromorphic families of compact operators. Arch. Rational Meeh. Anal. 31 (1969) 372-379. [43] P. Takak. A spectral mapping theorem for the exponential function in linear transport theory. Transp . Theory Stat. Phys. 14(5) (1985) 655-667. [44] S. ukai. Eigenvalues of the neutron transport operator for a homogeneous finite moderator. J. Math. Anal. Appl. 30 (1967) 297-314. [45] S. Ukai. Solutions of the Boltzmann equation, in Patterns and wavesQualitative analysis of nonlinear differential equations, pp. 37-96. Studies. Math. Appl. 18, 1986. [46] I. Vidav. Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22 (1968) 144-155. [47] I. Vidav. Spectra of perturbed semigoups with applications to transport theory. J. Math. Anal. Appl. 30 (1970) 264-279. [48] V.S. Vladimirov. Mathematical Problems in the One-velocity Theory of Particle Transport. Atomic Energy of Canada. Ltd. Chalk lliver. Ont Report. AECL-1661 (1963). [49] J. Voigt. Positivity in time dependent linear transport theory. Acta. Appl. Math. 2 (1984) 311-331. [50] J . Voigt. Spectral properties of the neutron transport equation. J . Math. Anal. Appl. 106 (1985) 140-153. [51] J. Voigt. Functional analytic treatment of the initial boundary value problem for collisionless gases. Habilitationschrijt, Universitiit Miinchen, 1981. [52] L. Weis. A Generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory. J. Math. Anal. Appl. 129 (1988) 6-23. [53] K. Yosida. Functional Analysis. Springer Verlag, 1978.
Chapter 5
On the leading eigenelements of transport operators 5.1
Introduction
This chapter is devoted to a thorough analysis of the leading eigenelements of neutron transport operators. It is well known that they are the only eigenelements of physical significance and their importance in nuclear reactor theory (e.g. pulsed neutron experiments) motivated a great deal of the subsequent spectral literature in neutron transport theory. The existence of such leading eigenelements is strongly tied to positivity. The role of positivity in nuclear reactor theory was emphasized very early by G. Birkhoff [7] [8] [9] and G.J. Habetler and M.A. Martino [21]. Basic results on the leading eigenelements of transport operators were given by I. Vidav [45], J. Mika [32], T. Hiraoka and S. Uka'i [22], N. Angelescu and V. Protopopescu [3] and other developments followed. On the other hand, independently of transport theory, the spectral theory of positive operators and positive semigroups developed in its own right to a high degree of refinement and now very general functional analytic results are available which can be found , for instance, in the monograph by R. Nagel et al [38]. In Section 5.2, we recall some of the basic results on the spectral theory of positive operators. The aim is not, of course, to exhaust the theory but simply to select those results which present a great interest for neutron transport problems. Besides relatively well-known results on the peripheral spectrum, we will 99
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Topics in Neutron llansport Theory
present more specialized results such as strict comparison of spectral radii by I. Marek [30],the superconvexity result by J.F.C. Kingman [27] and T. Kato [26], B.de. Pagter's result on irreducible compact operators [40] and related results. We will show, in the subsequent sections, how such tools illuminate the properties of the boundary spectrum of transport operators. For a quick introduction to basic definitions and spectral results on positive semigroups on Banach lattices we refer to Ph. Clement et a1 [15]. A more systematic account is given in R. Nagel et a1 [38]. In Section 5.3, we present several approaches of the irreducibility of transport semigroups giving different points of view on this problem. In Section 5.4, we give very general existence results for eigenvalues, when the velocity space is bounded away from zero. Such results rely on the compactness results of Chapter 4 and on irreducibility arguments. In Section 5.5, we derive a spectral link between the transport operator and its bounded part. This link provides us with nonexzstence results we give in Section 5.6. We also give, in this section, an upper bound of the leading eigenvalue in terms of the spectral radius of the collision operator, when the velocity space is bounded away from zero. In Section 5.7, we study the case where the velocity space is not bounded away from zero and give several existence results for eigenvalues, based on estimates from below of the spectral radii of suitable operators. Section 5.8 is devoted to strict monotonicity results of the leading eigenvalue with respect to the cross sections or the spatial domain. In Section 5.9, we give a result on the domain derivative (with respect to vector fields) of the leading eigenvalue for a model transport operator and a simple formula for the derivative. We sketch, in Section 5.10, an approximation theory (with error estimates) of the leading eigenelements of transport operators, based on a projection method. The criticality eigenvalue problem is dealt with in Section 5.11. Finally, Section 5.12 is devoted to the extension of (some of) the previous results to transport equations with delayed neutrons.
5.2
Spectral properties of positive operators
Let X be complex Banach lattice. We denote by X+ the cone of positive elements. The notation x > 0 means x E X+ and x # 0. We also define the dual cone by X' - x ' E x ' ; ( x ' , x ) ~ o V x e X +
+-I
1
where X' is the dual space and (., .) is the duality pairing. In the Lp(R; dp) spaces those definitions correspond to the usual notion of non-negative functions. An operator 0 E L(X) is said to be positive if it leaves the positive cone invariant. We recall that a co-semigroup {U(t); t 2 0) on X with gen-
101
Chapter 5. On the leading eigenelements
erator T is said to be positive if each operator U (t) is positive. We point out the following useful characterization Proposition 5.1 ([15] p . 161). {U(t); t ~ O} is positive if and only if (AT)-l is positive for some A > w, where W is the type of {U(t); t ~ O}.
For a general CO-sernigroup {U(t); t that (A - T)-l =
~
O} in a Banach space, it is known
l~o e->.tU(t)dt
where W is the type of {U(t); t convergent . We recall that
~
; ReA>
(5.1)
W
O} . The integral (5.1) is absolutely norm
s(T):= sup ReA::;
W
>'E'tU(t)dt ; ReA> seT)
(5.2)
where (5.2) exists as a norm convergent improper integral.
The concept of irreducibility is probably one of the most inIportant ones in the theory of positive operators. To introduce it, we recall that a subspace Y of X is said to be an ideal if Ixl ::; Iyl and y E Y inIply x E Y where II denotes the absolute value. In the V(n ;dJ-L) (1::; p < 00) spaces, the ideals are of the form
{c,o
E
£P(n; dJ-L); c,o = 0 on A} for some measurable subset A
c n.
A positive operator 0 E L(X) is said to be irreducible if there is no closed ideal (except X and {O}) which is invariant under O. The irreducibility can be characterized in the following way. Proposition 5.3 A positive operator 0 E L(X) is irreducible if for any x > 0 and Xl > 0 there exists an integer n (depending possibly on x and Xl) / such that (onx,x ) > O.
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An element x E X+ is called quasi-interior if (x', x) > 0 for any x' > o. A functional x' E X~ is called strictly positive if (x', x) > 0 for any x > o. In the V(n; dft) spaces, quasi-interior elements correspond to strictly positive almost everywhere functions. A positive operator 0 E L(X) is called strongly irreducible (or positivity improving) if Ox is quasi-interior for any x > O. We note that 0 is irreducible if some power of 0 is positivity improving. This provides us with a practical mean to check the irreducibility. A positive CO-semigroup {U(t); t ~ O} on X is said to be irreducible if there is no closed ideal (except X and {O} which is invariant under U (t) for all t ~ O. We point out that a sufficient (but not necessary) condition for a positive CO-semigroup {U(t); t ~ O} to be irreducible is that, for some to > 0, the operator U(to) be irreducible. The following characterization proves useful for the sequel
r
Proposition 5.4 ([15] p. 165). Let {U(t); t ~ O} be a positive co-semigroup on X with generator T. Then the following assertions are equivalent: (i) {U(t); t ~ O} is irreducible. (ii) For every x> 0, x' > 0 there exists t ~ 0 such that (U(t)x, x') > O. (iii) (A - T)-l is strongly irreducible for all (for some) A > s(T). (iiii) (A - T)-l is irreducible for all (for some) A > s(T). We are now in a position to give the main spectral results. We begin with the most important one Theorem 5.1 ([41]). Let 0 E L(X) be a positive operator. Then ra(O) E a(O) . The analog for generators of positive semigroups is the following Theorem 5.2 ([15] p. 202). Let {U(t); t ~ O} be a positive co-semigroup on X with generator T. Then the spectral bound of T s(T)
=
sup ReA E a(T) AEa(T)
provided that a(T)
=1=
0.
The peripheral spectrum of a positive operator 0 E L(X) is defined by
{A E a(O); IAI = ra(O)} while the peripheral (or rather the boundary) spectrum of the generator T of a positive co-semigroup is defined by a+(T) = {A E a(T); ReA = s(T)} .
The peripheral spectrum enjoys nice properties
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Chapter 5. On the leading eigenelements
Theorem 5.3 ([15] pp. 202 and 205). Let {U(t)j t 2: O} be a positive cosemigroup on X with generatorT. If s(T) is a pole of the resolvent ().._T)-l of order p, then any other pole J1, E O'+(T) is of order::; p. Moreover O'+(T) is cyclic, i. e. ).. E 0'+ (T) implies s(T) + ikhn)" E 0'+ (T) for every k E Z . More precise pictures of O'+(T) hold under additional assumptions
Theorem 5.4 ([15] p. 209). Let {U(t)j t 2: O} be an irreducible co-semigroup on X with generator T. Then there exists II 2: 0 such that O'+(T) = s(T)
+ illZ
and the elements of O'+(T) are first order poles with algebraic multiplicity one. Moreover, there exists a quasi-interior Xo E X+ such that Txo s(T)xo and a strictly positive x~ E X~ such that T' x~ = s(T)x~. Replacing the irreducibility condition by an assumption on the essential type yields
Theorem 5.5 Let {U(t)j t 2: O} be a positive co-semigroup on X with generator T. We assume that We < W where wand We are respectively the type and the essential type of {U(t)j t 2: O} . Then
Furthermore, there exists c: > 0 such that O'(T) n {Aj Re).. 2: W i.e.
W
-
c:}
= {w}
is a strictly dominant eigenvalue ofT.
Proof: The result follows easily from the fact that O'(T) n {>.; Re).. 2: Q} is finite for We < Q < W and from the cyclicity of the boundary spectrum (Theorem 5.3) . We mention a useful tool to check the irreducibility for positively perturbed positive CO-semigroups which can be used for transport problems
Theorem 5.6 ([38] p . 307). Let {U(t)j t 2: O} be a positive co-semigroup on X with generator T and let K E L(X) be positive. Let {V(t)j t 2: O} be the co-semigroup generated by T + K. Let I c X be a closed ideal. Then the following assertions are equivalent (i) I is invariant under {V(t)j t 2: O} . (ii) I is invariant under both {U(t)j t 2: O} and K.
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An important question of pure and applied interest is under which conditions is the spectral radius of a positive operator strictly positive ? The known criteria (up to 1983) (including Ando-Krieger's theorem for integral operators) tied to the concept of irreducibility are summarized in H.H. Schaefer ([42] Theorem A). We point out an extension of Ando-Krieger's theorem to ordered Banach spaces by V. Caselles [12] . We mention here a general abstract result by B.de. Pagter [40] (published in 1986) which will be used quite often in the sequel Theorem 5.7 Let 0 E L(X) be a compact irreducible compact operator. Then ru(O) > O.
The following result, due essentially to H.H. Schaefer [42] , will prove useful for transport problems Theorem 5.8 Let {U(t) j t ~ O} be an irreducible eo-semigroup on a Banach lattice X with generator T If U(t) is compact for t large enough then
a(T)
-10.
Actually this result is not stated in Schaefer's paper [42] . However its proof is exactly the same as that of the case X = L1(p,) ([42] Theorem B (iii)) by using B.de. Pagter's result [40]. The previous result provides us with the existence of an eigenvalue for generators under certain conditions on the semigroups. We will give thereafter (Theorem 5.12) another approach (for perturbed semigroups) tied to properties of the generator and also well suited to transport problems. To this end we introduce the concept of superconvexity and its applications to positive operators depending on a real parameter. This concept also plays a basic role for an approximation theory of the leading eigenelements of transport operators [13] (see also Section 5.10) . Definition 5.1 Let I be an interval of R . A function f : I superconvex if Log f(x) is a convex function.
-t
R+ is called
We first mention a result on the superconvexity of the spectral radius of positive matrices by J .F .C. Kingman [27] Theorem 5.9 Let A(h) = (ai ,j(h))~j=1 be a positive matrix depending on a parameter h E I . If ai,j (h) is superconvex for each i , j E {I, .. ., n}, then
h is superconvex.
-t
ru(A(h))
105
Chapter 5. On the leading eigenelements
We present now an infinite dimensional version of J.F.C. Kingman's result due to T. Kato [26] . To this end we recall the meaning of the superconvexity for vector valued mappings. Let A(h) be a family (indexed by hE 1) of operators acting on a Banach lattice X with positive cone X+. A function
f: hE I
-->
f(h) E X+
is called superconvex if, for each E: > 0 and each triplet hI ::; ho ::; h2 , there are finitely many Xj E X+ and real superconvex functions cpj(h) such that
f(hk) -
L CPj(hk)Xj
::;
E:
;
k = 0, 1,2.
j
We say that
A: hE 1--> A(h) E L(X) ; A(h)
~
0
is superconvex if, for each x E X+, hE 1--> A(h)x E X+ is super convex. We state T. Kato's result [26]
Theorem 5.10 If A(h) is superconvex in h, so is rO'(A(h)). Actually for the purpose of Theorem 5.12 we give a particular and simpler version of T. Kato's result based on the more tractable concept of weak superconvexity. A family A(h) E L(X) ; A(h) ~ 0 is called weakly superconvex if the real valued function (A(h)x, x'} is superconvex for each x E X+ and x' E X~ . Then we have the following result which can be deduced directly from J.F.C. Kingman's result by approximation:
Theorem 5.11 ([13]). Let A(h) E L(X); A(h) ~ 0 be a weakly superconvex family of compact operators. Then rO'(A(h)) is superconvex. We are in a position to prove a result complementing Theorem 5.8:
Theorem 5.12 Let {U(t); t ~ O} be a positive co-semigroup on a Banach lattice X with generator T. Let there exist a > 0 such that U(t) = 0 for t > a. Let B E L(X) be a positive operator and {V(t); t ~ O} be the cosemigroup generated by T + B. If some power [(>' - T)-I B]m (>. E R) is compact and irreducible then a(T + B) i- 0.
106
Topics in Neutron lkansport Theory
Proof: We give two proofs of this results. We observe that (A - T)-' is entire since {U(t);t 0) is nilpotent. The first proof is based on the assumption (different from that of the statement) that (A-T) -' [B(X - T) -'1" is compact and irreducible for some integer m. The second proof (corresponding to the statement) relies on Kingman-Kato's convexity result. Let s(T B) be the spectral bound of T B
>
+
+
It is known (see [38] Proposition 2.5, p. 67) that, for X r, [(A - T -
B)-'1
=
1 dist(X;u(T
+
T)-'1
1
-
+ B)) - X - s(T + B)
so that u(T B) # 0 if and only if r, [(A - T hand, according to Lemma 8.1, r, [B(X-
> s(T + B),
B)-'1
> 0. On the other
< 1 for X > s(T + B)
and
Hence
from Theorem 5.7 and this ends the first proof. We give now a differenta p proach. By analyticity arguments, [(A - T ) - ' B ] ~is compact for all X and, by Gohberg-Shmulyan's theorem, u(T B) consists, at most, of isolated eigenvalues with finite algebraic multiplicities. In view of Theorem 5.2, u(T + B) # 0 is equivalent to the existence of a real eigenvalue. According to Theorem 5.7 and the spectral mapping theorem, r,((X - T)-'B) > 0 is an eigenvalue of (A - T)-'B associated with a positive eigenfunction. On the other hand, X is an eigenvalue of T B if and only if 1 is an eigenvalue of (A - T)-'B. Hence the problem amounts to the existence of X E R such that r,((X - T)-'B) = 1. By analyticity arguments,
+
+
X E R + r,((X
- T)-'B)
whence the problem amounts to
is strictly decreasing
107
Chapter 5. On the leading eigenelements We start with
Set 0(>')
= [(>. - T)-1 B]m >. E R
---->
and let us show that
0(>') is weakly superconvex,
i.e. for all x> 0 and x' > 0,
is superconvex. The integrand is superconvex. Since linear combinations with positive coefficients and limits of superconvex functions are superconvex ([26] Lemma 2.2), then (5.4) follows by approximating the integral by finite sums and passing to the limit. According to Theorem 5.11,
>. E R
---->
ru(O(>'))
is super convex and hence convex. We note that ru(O(>')) is analytic in >. because ru(O(>')) is a simple eigenvalue of the analytic operator 0(>') (see T. Kato [25]). Finally, as a differentiable strictly decreasing convex function, lim ru(O(>')) = .:\.-+-00
+00
so (5 .3) is satisfied. Another important question of pure and applied interest concerns the strict comparison of the spectral radii of two operators in a context of domination. Such questions were investigated a long time ago by 1. Marek [30]. For more recent results and references we refer to V. Caselles [11] . We quote here a very particular version of a result by 1. Marek [30] which will be used later to study the strict monotonicity of the leading eigenvalue of transport operators with respect to various parameters. Theorem 5.13 Let 0 1 and 02 be two positive bounded operators such that
and let 02 be power compact and irreducible. Then r u (01)
< ru(02) .
108
Topics in Neutron ~ m s p o r Theory t
The irreducibility of transport semigroups
5.3
Let R be a smooth open subset of Rn and let dp be a positive Radon measure on Rn such that dp{O) = 0. We denote by V the support of dp and refer to V as the velocity space. We consider neutron transport equations of the form
a f v.-a f at ax
+
+ ~ ( xv ),f ( x ,v , t )
J
=
k ( z ,v , v ' )f
( 2 ,v', t ) d P ( v t )
where
r- = { ( x , ~E)dR x
V ; v.n(x) < 0 )
and n ( x ) is the outward normal at x E aR. Here, u ( x , v ) denotes the collision frequency at x for neutrons with velocity v , while k(., ., .) is the scattering kernel. We recall the streaming operator
T f = -v.-
af - ~ ( xv ),f ( x ,v ) ; f E D ( T )
ax
with domain
f E L p ( R x V ) ; v.-a f E Lp(R x V ) ,fir- = O
ax
where
Lp(R x V ) = L p ( R x V ; d x d p ( v ) ) ; 1 I p < oo. We assume, as usual, that the collision frequency is non-negative and bounded and that the scattering kernel is non-negative and defines a bounded operator K on P ( R x V). Note that T generates the following Q-semigroup
where
s(x,v)=inf{s>O;x-sv$R). This section is devoted to the irreducibility of the %-semigroup { V ( t ); t 2 0 ) with generator T K. Different approaches of the irreducibility are given. We begin with a result by J. Voigt [46] concerning the continuow models.
+
109
Chapter 5. On the leading eigenelements
We first recall the notion of path diameter p-diam( n), for an open connected n c Rn. For x, x' E n, we define the path distance p-dist(x, x') = inf {length of C; C polygonal path connecting x and x'} p-diam(n) = sup {p-dist(x, x'); x, x'
En} .
Theorem 5.14 Let n be connected and let VeRn (open) and df-£(v) the restriction of the Lebesgue measure to V . Let there exist 0 ~ Cl < C2 ~ 00 such that
and k(x, v, v') > 0 a.e. on (n x Vo x V) u (n x V x Vo)
(5.7)
then {Vet); t ;::: O} is irreducible. Moreover, if C2 = 00 then {Vet); t ;::: O} is positivity improving for all t > O. If C2 < 00 and p-diam(n)< 00 then {Vet); t ;::: O} is positivity improving for all t > P-di~:,,(n). The proof, of geometrical character, is based on Proposition 5.4 part (ii) and the analysis of the second term of the Dyson-Phillips expansion of {Vet); t ;::: O} from {U(t); t ;::: O}. We present now a second approach of the irreducibility based on the resolvent characterization (Proposition 5.4 part (iii)) . Theorem 5.15 Let there exists an integerm such that [()._T)-lK]m is positivity improving, then {Vet); t ;::: O} is irreducible. This condition is satisfied (with m = 2) by the continuous models under the assumption (5.7) and the additional assumption that n be convex.
Proof: For)' large enough, 00
(). - T - K)-l =
L [(). - T)-l K]i (). - T)-l o
and then (). - T - K)-l is irreducible if some power of (). - T)-l K is positivity improving. For the Lebesgue measure on Rn, K()' - T)-lK is an integral operator whose kernel is greater than or equal to
x(x-x
I
' I <sex,
x- x 10 e- ().+-( a.e.
"t
t'
t
On the other hand, I
1:=:/1))=1
X(lx-X'I<s(x,
if and only if n is convex. This ends the proof.
VX,X/En
0
Remark 5.1 We point out that the remainder terms of the Dyson-Phillips through the Laplace expansion are related to the powers [(A - T)-l transform so that it is not surprising that similar assumptions on the scattering kernel appear in both approaches. However, similar arguments led G. Greiner [18] [19] to a quite different criterion: k(x, v, v') > 0 in a neighborhood of the boundary
Kf
an.
Theorem 5.15 admits a multigroup version we give now. To this end it is convenient to recall the generation theory in this special context. The multigroup transport equation has the following form (5.8)
subject to initial and boundary conditions
where Here 1f;i(X, v, t), with (x, v, t) E particles with speed Ci. Finally
fi
n x Vi
= {(x,v) E
x R+, denotes the distribution of
an x Vi ;v.n(x) < O} .
We write (5.8) formally as a Cauchy problem d1f;
di"= T1f; + K1f;
; 1f;(O) = 1f;o
111
Chapter 5. On the leading eigenelements
T-
0 T2 0 0
C' 0 0 0
0 0 0
0 0 0 Tm
)
at.p ; Tit.p = -v. ax - (Ji(X, v)t.p(x, v) , t.p E D(Ti)
and
The entries of the matrix collision operator K = {Ki,j} (1 :s; i,j ::; m) consist of integral operators (with respect to velocities) with kernels ki ,j(x, v, v') ~ O. We note that Ti generates the eo-semigroup
Ui(t)t.p = e
-it
u,(x-rv,v)dr
t.p(x - tv, v)x(t
0
< sex, v)) ; t.p
E p(n x Vi).
Thus the diagonal operator T with domain D(T) =
II
D(Ti)
l$i$m
generates a eo-semigroup {U(t); t ~ O} on
ITl$i$m
£P(n x Vi) where
Assuming that Ki,j E L(LP(n x Vj), p(n x Vi)) ; (1::; i,j
:s; m) ,
it follows that the Cauchy problem (5.8) is well-posed and governed by a multigroup transport semigroup {Vet); t ~ O} . We are now ready to give an irreducibility criterion for the multigroup transport semigroup. Theorem 5.16 Let n be convex. We assume that, for each 1 ::; i,j ::; m, there exists 1 :s; e :s; m such that ki,e(x,v,v')
> 0 on n x Vi
X
Ve ; ke,j(x,v',v)
> 0 on n x Ve x Vj (5.9)
then the multigroup transport semigroup is irreducible.
112
Topics in Neutron Transport Theory
Proof A calculation shows that the entries of the matrix K()" - T)-l K are given by [K(). - T)-l KL,j =
L
Ki,e(). - Te)-l Ke,j
l:5e:5m
and that Ki,e(). - Te)-l Ke,j E L(V(n x Vj), V(n x Vi)) is an integral operator with kernel greater than or equal to Ni,e,j (x , x' , V , v")
I
x-x )ke J. ( X = Cen-2ki e ( x, v, ce f:-:7T' I ,
IX-X
J
I
I"
, ce
x- x ,v f:-:7T' I
)
IX- X
where O'e(v) = infxEf! ae(x, v) . Assumption (5.9) is sufficient for .. [K()' - T)-l K] ',J
to be positivity improving and it follows that [(). - T)-l K]2 is also positivity improving. We conclude by using the abstract result given in Theorem 5.15. 0 Remark 5.2 The irreducibility of the transport semigroup may also be checked by means of Theorem 5.6. We refer to [38] p. 309, for the analysis of the one dimensional case.
5.4
A general existence result
We turn to the general operator we started with in the previous section A=T+K
where
8f Tf = -v . 8x - a(x, v)f(x, v)
f E D(T)
with domain D(T)
and Krp =
=
{f E V(n x V;dxdJ.L(v));
J
v.:~
E
LP, fw- =
o}
k(x,v,v')rp(x ,v' )dJ.L(v') ; rp E V(n x V;dxdJ.L(v)).
Chapter 5. On the leading eigenelements
113
We prove now the existence of an eigenvalue for T + K, when the velocity space is bounded away from zero, under very general assumptions. In such a case, the unperturbed semigroup {U(t); t ~ O} is nilpotent, i.e. U(t) = 0 for t > to = ..4. where d is the diameter of n and Vo is the minimum speed. Vo We give two general existence results based respectively on two different functional analytic arguments (Theorem 5.8 and Theorem 5.12). Theorem 5.17 Let 0 rf. V and the transport semigroup {V(t); t ~ O} be irreducible. We assume that the collision operator is regular (in the sense of Definition 4.1) and that n is bounded. If 1 < p < 00 and if the Radon measure d/l is such that translated hyperplanes have zero d/l- measure, then a(T+K) i- 0. If p = 1 and if the Radon measure d/l satisfies the geometrical condition (4.31), then a(T + K) i- 0.
Proof: When 1 < p < 00 then, according to Theorem 4.8, the remainder R3 (t) of the Dyson-Phillips expansion is compact in V (n xV; dxd/l( v)) . When p = 1 then, according to Theorem 4.12, R3(t) is weakly compact in Ll(n x V; dxd/l(v)).1t follows, from Corollary 2.1, that R7(t) is compact in Ll(n x V; dxd/l(v)). Hence, in both cases, some remainder term is compact. On the other hand, U(t) = 0 for t > ~ = to and a simple calculation shows that V(t) = Rm(t) for t > mto. Thus V(t) is compact for t large enough and the result follows from Theorem 5.8. 0 Remark 5.3 It is possible to improve further the previous result. The irreducibility relies on some strict positivity assumption on the scattering kernel as is shown in the previous section. Actually, it suffices that, for some ball Ben, the transport semigroup corresponding to the ball be irreducible since it is easy to see that the leading eigenvalue increases with the domain. Thus, in the context of continuous (resp. multigroup) models, it suffices that the condition (5.7) (resp. (5.9)) be satisfied in B instead of
n. Theorem 5.18 Let 0 rf. V and let n be bounded. We assume that the collision operator is regular (in the sense of Definition 4.1). (i) Let 1 < p < 00 and assume that d/l is such that the hyperplanes have zero d/l-measure. If K().. - T)-l is irreducible then a(T + K) i- 0. (ii) Let p = 1 and assume that d/l satisfies the geometrical condition (4 .9) . If [K().. - T)-1]2 is irreducible then a(T + K) i- 0.
Proof: According to Theorem 4.1 (resp. Theorem 4.4), K()" - T)-l (resp. [K()" - T)-1]2) is compact in V(n x V;dxd/l(v)) (resp. in Ll(n x V; dxd/l(v))) and the proof follows from Theorem 5.12. 0
Topics in Neutron Transport Theory
114
Remark 5.4 In the context of continuous or multigroup models (and if 0 is convex), the conditions (5.7) or (5.9) ensure the irreducibility of [K(A - T)-l]2 and thus the conclusion of Theorem 5.18. As pointed out in Remark 5.3, it suffices that such conditions be satisfied in a ball B c O.
5.5
A spectral inequality
We consider, in this section, the case where the velocity space is not bounded away from zero. We will assume that the collision frequency, the scattering kernel are homogeneous and that lim inf a(v) v-+O
= vEV inf a(v) = 'T} .
We are going to show a useful link between the spectrum of the transport operator and that of its bounded part B : r.p E P(Vj dJL(v))
~ -a(v)r.p(v) +
J
k(v, v')r.p(v')dJL(v') E P(Vj dJL(v)).
We begin with the following observation Proposition 5.5 Let K be power compact in V(V; dJL(v)) . Then aas(B) := a(B)
n p;ReA >
-'T}}
consists of, at most, isolated eigenvalues with finite algebraic multiplicities. Moreover if aas(B) =/:- 0 then there exists a leading eigenvalue :X. Proof: The resolvent of the multiplication operator r.p E P(V; dJL(v)) ~ -a(v)r.p(v)
is given by r.p(v) r.p E LP(V; dJL(v)) ~ A + a(v) = (A
+ a)
-1
r.p ; ReA>
-'T}.
A!
For real A, K(A + a)-l is dominated by K and is thus power compact. This implies the first part of the propositiori. The rest follows from general results on positive operators, for instance Theorem 5.2. We are in a position to state Theorem 5.19 Let aas(T + K) = aCT + K) n {A; ReA> -'T}} =/:- 0 and let A be the leading eigenvalue ofT + K. Then aas(B) =/:- 0 and A ::;:X. If K is irreducible in V(V ;dJL(v)) then A
-7] --+
ru(K(oo, A))
is continuous (because K(oo, A) is power compact and ru(K(oo, ,\)) is an eigenvalue), nonincreasing and tends to zero at infinity, there exists); 2: A such that ru(K(oo, );)) = l. Thus there exists a non-negative g such that K(oo, );)g = g ,
i.e.
1 =----Kg=g
A+cr(V)
which amounts to Bg = );g. In the case where K is irreducible in V(Vj dJl(v)) then K(oo, A) is also irreducible and, thanks to Theorem 5.13, (5.12) yields ru(K(oo, A)) > ru(K(d, A)) 2: 1
which implies that A < );.
5.6
0
Nonexistence results
We give some consequences of the previous section. We begin with Corollary 5.1 Let K be quasinilpotent (i.e. ru(K) = 0) thenaas(T+K) =
o regardless of the size of n.
Proof: Indeed, we have K(oo,);) ~ >:!ryK and consequently the equality ru(K(oo, );)) = 1 is not possible.
0
Remark 5.5 If the scattering kernel is such that k(v, Vi) = 0 for
Ivl 2:
Iv'l
then ru(K) = O. It is possible to exploit Theorem 5.19 to derive nonexistence results of a different kind. For instance
Theorem 5.20 Let p = 1 and let a(v) >
7]
a.e. We assume that
k(v, Vi) dJl( v) ~ l. a () v -7]
J
Then aas(T + K) = 0 regardless of the size of n.
(5.13)
117
Chapter 5. On the leading eigenelements
Proof: It suffices to show that (5.13) implies that <Jas(B) = 0. Suppose the contrary and let>: be the leading eigenvalue of B. Thus 1 A + <J( v)
J ' , , -
1 k(v, v )rp(v )dll(V ) = rp ; A> -1/ , rp E L+(V ; dll(V )) , rp
f:: O.
By integrating over V we get
Hence
which ends the proof. Remark 5.6 If 1/ > 0 then similar calculations show that T subcritical (i .e. <Jas(T + K)
c {ReA < O} V D)
sufficient condition of subcriticality in L2 is /
if
J
+K
is always
kS'(':;)dll(V) :S 1. A
Ja~~);;l~)dll(V)dll(V') :s:
1.
We give another remark Remark 5.7 Eigenvalues cannot exist if the velocity space V does not contain at least a subset symmetric with respect to the origin. Indeed, if V is a half ball then ra((A - T)-l K) = 0 (see [36]). Although unphysical, this example shows that the converse to Theorem 5.1 9 is not true in general since the spectral theory of B is not related to the geometry of the velocity space v.
We mention now the classical disappearance phenomenon for small bodies Theorem 5.21 Let K be the closed operator in V(V; dll(V)) K: rp E D(K) { D(K)
-->
Krp =
Ivl- 1 Krp
= {rp E V(V; dll(V)); Ivl- 1 K rp E V(V ; dll(V)) } .
If D(K) = V(V; dll(V)) then <Jas(T+K) diameter of D.
= 0 for d IIKII :S 1 where d is the
Topics in Neutron Transport Theory
118 Proof: We note that
(A - T)-l'tjJ =
Ivl
_
( S(X,w)
1
io
_
e
v
( A+U ( V )) '
Ivi
'tjJ(x - sw, v)ds ; w =
f0
and that
Illvl (A - T)-lIIL(LP(nXV;dXdJ.L(V») ~ d. Hence
II(A - T)-l KII < d IIKII ; ReA> -TJ
which ends the proof since 1 ~ ap((A - T)-lK) . 0 We end this section with an upper estimate of the leading eigenvalue when the velocity space is bounded away from zero. Theorem 5.22 Let 0 ~ V, let c be the minimum speed and d be the diameter of n. If a(T + K) #- 0 then the leading eigenvalue A of T + K satisfies the estimate d
la c e-(>.+infu(,))sds x ru(K) ~ 1.
(5.14)
Proof. Now, for any A, r(X,V)
(A - T)-lcp =
io
e-(>'+u(v»cp(x - sv , v)ds ; cp E LP(n x V; dxd{t(v)).
By using the calculations in Section 5.5, (5.11) shows that d
'tjJ(v)
~ J~VI ~
J:
e-(>'+u(v»ds
J k(v,v')'tjJ(v')d{t(v')
d
e-(>.+infu(,»sds x K'tjJ =: K'tjJ
so that ru(K) ~ 1 and this proves the claim.
5.7
0
Existence results
We deal, here, with the case where the velocity space is not bounded away from zero. We also assume that the collision frequency, the scattering kernel are homogeneous and that lim inf a(v) = inf a(v) = TJ . V""" 0
vEV
119
Chapter 5. On the leading eigenelements We assume that
(>. - T)-l K is power compact in V(n x Vj dxd/L(v)) .
(5.15)
Conditions on K implying (5.15) are given in Chapter 4. To fix the ideas we will assume that K is compact on V(Vj d/L( v)) (1 < p < (0) or dominated by a compact operator in L1(Vjd/L(v)). If (>. - T)-lK is irreducible then Theorem 5.7 ensures that
ra«>' - T)-l K) > O. By analyticity arguments (see Gohberg-Shmulyan's theorem [43]) ,
>.
]-7], oo[
E
-+
ra«>' - T)-l K)
is strictly decreasing (and continuous). It follows that
if and only if lim ra«>' - T)-l K) > 1
>'-+-'7
and then the leading eigenvalue>: of T
+K
is characterized by the equality
Thus, we are faced with the problem of obtaining explicit lower bounds of the spectral radius ra«>' - T)-l K) or, which amounts to the same, explicit lower bounds of ra(K(>. - T)-l) . We set r(X,V')
a(x , v', >.)
= Jo
,
e-(A+a(v))sds
j
>. 2: -7].
n x ]-7],00[, we define the operator
For each (x, >.) E
H(x, >.) : t.p
E
V(Vj d/L(v))
-+
J
a(x, v' , >')k(v, v')t.p(v')d/L(v').
It is easy to see that H( x, >.) inherits the compactness assumptions of K . We are going to define a parameter T(>') playing the role of a "uniform spectral radius" of H(x, >.), i.e. independent of x E n. We proceed as follows. Let
X(>.) =
b
2: OJ :3 t.p
E L~(V), t.p
=f. 0, H(x, >.)t.p 2: "(t.p
and
T(>') = sup bE X(>.)} . It is easy to see that T(>') is nonincreasing.
\;f
xE
n}
Topics in Neutron Transport Theory
120
Theorem 5.23 r,(K(X - T)-l) 2 r(X). Moreover, if lim r(X)
A+-q
>1
Proof: The result is trivial when r(X) = 0. Assume that r(X) let y > 0, y E X(X). There exists cp E L:(V) such that /a(.,
v', X)k(v, d)cp(d)dp(d)
> 0 and
> 79 , V x E R.
Then $(x, v) = cp(v) E LP(R x V; dxdp(v)) and
whence ru(K(X - T)-')
L7
and this ends the proof since y E X(X) is arbitrary. 0 The interest of Theorem 5.23 resides in the fact that r(X) can be estimated. We illustrate this by the following examples. We assume that dp(v) = dv V={v;O o}
such that
,
I
SeX, v) > 0 ;
vTJI E II(x).
The hemisphere TI(x) is the set of unit velocities pointing outside 0. at x E an. We set sn-l if x E 0. II(x) = { TI(x) if x E an. Thus
I I
sex , v) > 0 ;
TJI E II(x) , x E 0.. V
-
(5.18)
The basic consequence of (5.18) is that 't/ x
En,
a(x ,. , A) > 0 on a subset of V of positive measure.
Corollary 5.2 Let G(v) =
-rCA) Moreover: (i) If ~£I)
~ xinf Efl E
inf vEv
k(v, v' ) > 0 a.e. Then
J
G(v')a(x, v', A)dv' = leA) > 0 ; A> -T]
Ll(V) then lim>'-+-'7 I(A) =
infxEfl
J G(v')a(x, v', -T])dv'.
(ii) If for any solid cone C with vertex zero ~£I) fj. Ll(C) , then lim leA) =
>'-+-'7
+00.
Proof: Let cp(v) = 1. Then H(x , A)cp
I ~J
=
a(x , v' , A)k(v, v')cp(v')dv'
G(v' )a(x, v' , A)dv'
~ leA) =
I(A)cp
whence -rCA) ~ leA) . It follows, from (5.17) and (5.19) , that
x
(5.19)
En
--t
lex, A) =
J
G(v' )a(x, v', A)dv'
122
Topics in Neutron Transport Theory
is continuous and strictly positive, so that l(A) is continuous and for any x En; l(x, A)
--+
> O. In the case (i), l(., -T])
l(x, -T]) as A
--+
-T].
Using a decreasing sequence Aj --+ -T] and Dini's theorem, l(x, Aj) l(x, -T]) uniformly on and consequently
n
lim l(A) = inf
xEfl
)..-+-7)
f
--+
G(v')a(x, v', -T])dv' .
In the case (ii), by the monotone convergence theorem, l(x, Aj)
--+
f
(s(x ,v ' ) ,
dv' G(v') Jo
e-(O'(v )-7)sds
= l(x , -T]).
We note that l(x, -T]);::::
where w'
=
ful.
f
,,'
dv' G(v')
PTEI1(x)
f
S(X ,V ' ) ,
e-(0'(v)-7)sds 0
On the other hand, in view of (5.16),
(O'(v ' ) - T])
Iv'l
:S c
O. It follows
Vx E
-n.
Chapter 5. O n the leading eigenelements
&
123
a. Finally l ( x ,Aj) + oo uniformly on a
B y using Dini's theorem,
+0
uniformly o n
and limx,-, l(A) = oo.0 Similar arguments yield Corollary 5.3 Let k(., v ' ) be continuous on I/ and k ( v ,v ' )
T ( A ) >_
inf
I
(x,~)ERxV
Moreover: (i) I f k(u, v ' )
> 0 a.e. Then
k ( v ,v ' ) a ( x ,v ' , X)dvi = q A ) > 0 ; A > -?
< ~ ( u ' where )
E
L 1 ( V ) then
$ L 1 ( C ) for any solid cone C with vertex zero then
(ii) If
For degenerate scattering kernels we have a more precise result
zgl
Corollary 5.4 Let k ( v , v ' ) = f j ( v ) g j ( v ' )where f j E L?(V), S,. E LII+(V)(p-l 9-I = 1 ) . Let h i j ( v f )= g i ( ~ 'f )j ( v l ) and
+
mij(A)=inf
xEn
/
Let M ( X ) be the matrix { m i q (A); j 1 Proof: Let cp E LP(V), then
Let us look for cp 2 0 o f the form
hij(d)a(x,vl,~)dv'.
< i,j
5 N ) . Then r ( A ) 2 r,(M(X)).
Topics in Neutron Tkansport Theory
124
and y such that H ( x , X)cp 2 ycp for all x E R, i.e.
It suffices that
and clearly (5.20) is satisfied if
where ,B(X) is the vector {Pi(X)). TO solve (5.21) it sufEces to take y = T,(M(X)) and ,B(X) a non-negative eigenvector of M(X) associated with the eigenvalue y. 0
Remark 5.8 More precise exzstence results can be obtained by comparison arguments (see [36] Theorems 5 and 6).
5.8
Strict monotonicity properties of the leading eigenvalue
We are concerned in this section with the monotonicity dependence of the leading eigenvalue with respect to the parameters of the transport operator, i.e. the spatial domain, the collision frequency and the collision operator. Let R C Rn be smooth open and bounded. For each collision frequency a(.,.) E LY(R x V) and non-negative collision operator
we define the corresponding transport operator on IP(R x V; dxdp(v))
where
Chapter 5. On the leading eigenelements
125
with domain independent of a( ., .)
D(Tq)
={fEV(OXV;dXdJ.L(V)); v. ~~ EV, flL =o}.
We assume, as usual, that K(>.. - Tq)-1 is power compact on LP(O x V;dxdJ.L(v)). When aas(Tq + K) =I- 0, we denote by >"(Aq,K) the leading eigenvalue of Aq,K . Theorem 5.24 Let K and K be two collision operators such that
K 5: K ; K =I- K . If aas(Tq + K) =I- 0 then aas(Tq + K) =I- 0 and >"(Aq,K) 5: >"(Aq,K) · If K(>.. - T)-1 is irreducible then >"(Aq,K) < >"(Aq,K) . Proof: Let >"1 = >"(Aq,K), then rq(K(>"1 - Tq )-I) = 1.
On the other hand,
According to the first part of (5.22),
whence there exists >"2
~
>"1 such that rq(K(>"2 - Tq )-I) = 1
so that aas(Tq + K) =I- 0 and >"2 = >"(Aq K) . If K(>" - T)-1 is irreducible then (5.22) implies, according to Theore~ 5.13,
so that
rq(K(>"1 - Tq )-1) > 1 and therefore there exists >"2 > >"1 such that
which ends the proof. Similarly we have
Topics in Neutron lransport Theory
126
Theorem 5.25 Let al(.,.) and a2(., .) be two collision frequencies such that al(.,. ) 2:: a2(" ' ) j al(" ') i a2(" .).
We assume, in addition, that the spectral bounds ofTu; (i = 1,2) are the same. Ifaas(Tu1+K) i 0 thenaas (Tu2 +K) i 0 andA(Au1 ,K)::; A(A u2 ,K). If K(A - T ( 2)-1 is irreducible then A(Au1,K) < A(A u2 ,K) . Proof: The condition on the spectral bound is automatically satisfied if the velocity space is bounded away from zero. If 0 E V and if, for instance, the collision frequencies are homogeneous, this condition means
On the other hand,
for A greater than the spectral bound of Tu;. The first part of (5.23) implies
which proves the first claim. If K(A - T (2 )-1 is irreducible, then using Theorem 5.13, and we conclude as previously. We consider now the monotonicity with respect to the spatial domain. To this end we assume that the collision frequency a(.) and the collision operator K are homogeneous. Theorem 5.26 Let f!l and f!2 be smooth open and bounded subsets of Rn such that f!l C f!2 j f!l i f!2 '
Let Tl and T2 be respectively the streaming operator in V(f!l x Vj dxdl1(v)) and in V(f!2 x Vj dxdl1(v)} . If aas(Tl +K} i 0 then aas(T2 +K) i 0 and A(TI
+ K)
::; A(T2
If (A - T 2)-1 K is irreducible then A(TI
+ K) .
+ K) < A(T2 + K) .
127
Chapter 5. On the leading eigenelements
+K
Proof. Let >'1 be the leading eigenvalue of Tl corresponding eigenfunction. We have
and let
CPI ;:::
0 be the
(5.24) where
Sl(X,V)
= inf {s > 0iX -
sv tJ. fh}
S2(X,V)
= inf {s > OiX -
sv tJ.
Let
n2 }
By assumption
SI(X,V)
~
S2(X,V) i (x,v)
It follows from (5.24) that, on
=
nl
xV.
(5.25)
n2 x V, CPI
'l/JI
E
{
on
nl
o on n2 - nl .
is less than or equal to (5.26) where Xl is the indicator function of the set n l . We define, on V(n2 x V; dxdJ.l.( v)) , the operator
According to (5.26),
Where Xl is the multiplication operator by Xl (.). Moreover
(5.27) Thus, the first part of (5.27) yields
Topics in Neutron llansport Theory
128 and there exists X 2 2 X I such that
which proves the first claim. If ( A - T2)-lK is irreducible, then according to Theorem 5.13, r,((Xl - T ~ ) - ' K )> 1 and there exists X2
> X1
such that
which proves the second claim.
0
Remark 5.9 The irreducibility assumptions for the usual models are satisfied under positivity conditions on the scattering kernel of the form (5.7) or (5.9) when the spatial domain is convex.
5.9
Domain derivative of the leading eigenvalue
By using compactness arguments it is possible to prove, for homogeneous cross-sections, that cr,,(T + K ) depends "continuously" on the spatial domain R, where the convergence of domains is defined by
Rj + R when
xj(.)+ x(.) a.e.
as j
+ oo
where xj(.)and x(.) are the indicator functions of Rj and $2 (see [37]). In particular, the leading eigenvalue depends "continuously" on the spatial domain R. We present a recent result [14] on the differentiability of the leading eigenvalue with respect to the spatial domain for a model transport operator. Let R be a smooth bounded and wnvex open subset of Rn and
We consider the transport operator
129
Chapter 5. On the leading eigenelements
where a is a constant. It is known (see, for instance, [36]) that aas(Tn) ¥- 0 at least for f2 large enough (i.e. contains a ball of radius large enough). We denote by )'(f2) the leading eigenvalue. We are concerned with the differentiability of f2 ~ )'(f2). Let us explain the meaning of this concept. Let Cl be the space of bounded continuous vector fields 8:Rn~Rn
having bounded derivatives up to order two. Let f2 be such that aas(Tn) 0. Let f2l1 = (I + 8)f2.
¥-
By continuity of the leading eigenvalue with respect to the domain
aas(Tno) when 8 lies in a neighborhood of 0 in A: 8 E
¥- 0
Cl. Thus, the mapping
cl ~ )'(f2l1) E J =
]-a, oo[
is well-defined in the neighborhood of the origin. Definition 5.2 We say that the leading e'igenvalue ). is differentiable at f2 if A is Prechet differentiable at the origin. We denote by).' (f2, 8) the derivative in the direction 8, i.e. ),'(f2,8) = A'(O)(8) where A'(O) is the Frechet derivative of A at O. Let
fn
Let gn(x)
> 0 be the leading eigenfunction normalized by
k(i
= Iv fn(x, v)dv. E(TJ,x)
=
1
fn(x,v)dv)2dx = 1.
Finally let dt n n ; TJ E J , x E R
00
e-(U+'1)tt
Ixl
-
{O} .
Theorem 5.27 Let f2 be of class C2 and let its Gauss curvature be bounded below by a positive constant. Then). is differentiable at f2 and ).' (f2, 8)
=-
IL(~) Ian g~(x)8(x).n(x)ds(x) ;
8E
where n( x) is the outward normal at x E af2 and
1L(f2)
=
Jlnxn r
E aa ()'(f2), x - y)gn(x)gn(y)dxdy. TJ
cl
Topics in Neutron Ti-ansport Theory
130
We note that P ( R )< 0 so that the sign of X'(R,6) is given by that of
Estimates of ~ ( 0are) given in the following
Theorem 5.28 Under the previous conditions,
where d is the diameter of R and w, is the area of the unit sphere of Rn Assume that n = 2 or 3 and set
then
and
Remark 5.10 The assumption concerning the curvature is only intended to ensure that Re remains convex for small 6. The proofs being rather technical we refer the reader to [14].
5.10 An approximation theory of the leading eigenelements The aim of this section is to present an approximation theory with error estimates of the leading eigenelements of transport operators. We focus our attention only on the theoretical aspects of this method and refer to [5] for its practical implementation. We consider a transport operator with non-negative and homogeneous cross-sections
where
8f dx
Tf = - v - - ~ ( v ) f ( x , v )
; f E D(T)
131
Chapter 5. On the leading eigenelements with domain D(T) =
{f
K'P =
J
and
E LP(r! x
Vj dxdv)j v . ~~ E LP, flL = O}
k(v,v')'P(x,v')dv'
j
'P E LP(r! x Vjdxdv)
where r! is convex and
v=
{v E RnjO::;
We assume that aas(T + K)
i= 0,
Cl ::;
Ivl::; C2 < oo}.
i.e.
where -liminfv ..... oa(v) if 0 E V 'T/=
{
-00
otherwise.
The leading eigenvalue).' is determined by the equality
We assume also that
(oX - T)-l K is irreducible. We describe the approximation method. The spectral problem
is converted into In order to derive error estimates, it is more convenient to deal first with the smoother unknown 'P = K 1/J which is solution of (5.28)
and then 1/J is recovered by means of
We set
Topics in Neutron Transport Theory
132
The principle of the method consists in projecting (5.28) onto a finite dimensional subspace of piecewise constant functions. To this end we introduce a partition {Al", ... , A;;',,.} of n x V and the projection p",
7rm J(x,v) = LJiXAi"(x,v) ; (x,v)
E
nx V
i=l
where
1(Am) meas i
Jim =
1 "" Ai"
J(x ,v )dx dv
1:::; i
:::;Pm'
Let
df' =
diameter(Ar')
1:::; i
:::; Pm·
We assume that d m = max: {df'; 1 :::; i :::; Pm} -+ 0 as m -+ 00
which implies that 1Tm -+
I strongly in IJ' (n x V) .
Now we replace the operator H>. by the matrix
Hm(>\) = 7rmH>.7rm · It can be proved (see [13]) that Hm(>\) is irreducible and that
lim Pm().,)
m-+oo
= p().,)
uniformly on compact subsets of j7],00[
where so that lim r C1 (Hm ().,)) > 1 for m large enough.
>'-+-1)
Finally, we define
>:m as the solution of
and 0 is the eigenvalue we look for and 'P 2: 0 is the associated eigenfunction. Secondly, we have to "adjust the composition and the geometry of the physical system" in order that "( be equal to one. We are going to give a general answer to the first problem (5.29) . It is convenient to write it abstractly as 1
Tf+Ksf+-Kf'P=O; "(>0, 'P2:0.
(5.30)
"(
It is clear that a necessary condition to solve (5.30) is that seT) < O. Thus, (5.30) is clearly equivalent to (5.31) where
KC'Y) = Ks
1
+ -Kf · "(
We restrict ourselves to the Ll setting and assume that (0 - T)-1 KC'Y) is power compact and irreducible in Ll(n x V; dxdJ-L(v)). The irreducibility is satisfied if, for instance, (0 - T)-1 Ks is irreducible. Theorem 5.30 Under the preceding assumptions, the spectral problem (5.30) has a solution if and only if
Proof: In view of Theorem 5.7, p()..) = ra((O - T)-1 KC'Y))
> O.
Moreover, by analyticity arguments, p()..) is strictly decreasing in ).. (and continuous) so that the first part of (5.32) is necessary. We observe that (0 - T)-1 KC'Y) 2: (0 - T)-1 Ks
and, if K f -=I- 0, (0 - T)-1 KC'Y) -=I- (0 - T)-1 Ks ; V"( > O. Hence, in view of Theorem 5.13,
which shows that the second part of (5.32) is also necessary because nonnegative eigenfunctions of (0 - T)-1 KC'Y) are necessarily associated to its spectral radius (a consequence of the irreducibility assumption). Finally lim (0 - T)-1 KC'Y) = (0 - T)-1 Ks
-y->oo
in the operator norm
135
Chapter 5. On the leading eigenelements and ru((O - T)-l Ks) < 1 imply that
ru((O - T)-l K(r)) < 1 for 'Y large enough because of the upper semicontinuity of the spectral radius for the norm operator topology ([25]). Hence there exists a unique 'Y > 0 such that
ru((O - T)-l K('Y)) = 1 and this ends the proof. 0 We end this section by giving practical conditions on the cross-sections implying the assumptions of Theorem 5.30. We restrict ourselves to homogeneous cross-sections and suppose that lim inf O'(v) v ...... O
=
inf O'(v)
vEV
=",.
Proposition 5.6 (i) Let d be the diameter of 0, then
[1
II::;
II(O-T)-lKs
suPJ v' EV
-u(v).. 0 then lim ru((O - T)-l K(r))
1' ...... 0
= 00.
Proof The second part follows trivially from the inequality 1
ru((O - T)-l K(r)) ~ -ru((O - T)-l Kf). 'Y
We consider (i),
(O-T)-lKscp= so that
l
S (X'V)
e-u(v)sds
J
k s (v,v')cp(x-sv,v')dJ.1,(v')
J
1(0 - T)-lKscp(x, v)1 dx
n .. 0 (see [361).
5.12 The effects of delayed neutrons This section is devoted to the peripheral spectral theory of transport o p erators with delayed neutrons. We restrict ourselves to wntinuous models. The analysis of multigroup models is similar. We consider the system
and assume that k(. , ., .) ( 0 5 i _< m) are non-negative. It is clear that the semigroup { V ( t ) ;t 2 0) governing this system is positive since the unperturbed semigroup { U ( t ) ;t L 0 ) and the perturbation B are positive. We refer to Section 4.5 for the different notations and assumptions. We assume that
A1 1
A--A1
and the leading eigenvalue 5; is characterized by the equality
Note that - To)-'K(X))
L
A ~ ~ ~- TO)-'KI) ( ( A
A+A1
Thus it suEces that T , ( ( A - T ~ ) - ~ K ~>) 0. It is not difficult to see that the spectral radius of (A - To)-lKl as operator on P ( R x V) is greater than or equal t o its spectral radius as operator on P ( R o x V). The assumption of the theorem ensures the irreducibility of (A - To)-'K1 on P ( R o x V) (its square is positivity improving) so we conclude by Theorem 5.7. 0 We end this section by giving an irreducibility criterion of the semigroup {V(t); t L 0).
Theorem 5.32 We assume that
and, for any i E [1, ...,m ] , there exists K sure such that
C
V with positive Lebesgue mea-
then {V(t); t 2 0) is positivity improving (and hence irreducible) for t where p-diam(R) is the path diameter of Q. c2
>
Topics in Neutron llansport Theory
138
Proof: Let {Vo(t);t 2 0 ) be the semigroup on P ( R x V ) generated by To KO. According to Theorem 5.14, Assumption (5.39) ensures that { b ( t )t; 0 ) is positivity improving for t > f = . We note that system (5.35), (5.36) may be written as
+
>
i
f i ( x , v , t ) = e - X i t f i ( ~ ,+ ~ ) e - X i ( t - S ) ~ i f O ( ~ ;) d( 1~
o
+ EmXi a=1
I"
0
KJ (t - s )
Is
e-k(s-T) K j fo ( r ) d r .
0
Hence, for non-negative initial data and t
f ( v , t ) 1 Ki
>f,
fo(s)ds ; ( 1 5 i
Let Q0 = ( f O , f l , ...,f m ) # 0. By noting that each ( 1 i 5 m) is positivity improving, one sees that
-g(O) and consequently (6.8) has no solution if Ed llXll < 1. We point out that the IlHxll 5
1
same argument shows also that no complex eigenvalue exists. Conversely, suppose that I? is not bounded. Then, according to Lemma 6.2, the domain of the closed operator L a is not the whole space L 2 ( v ;d p ) and J consequently there exists cp E L 2 ( V ;d p ) with unit norm such that
J;;
$ L2 (v; dp)
153
Chapter 6. Form positive collision operators We define 'Ij; E £2(0 and
X
V) by 'Ij;(x, v)
= 101-t cp(v). Then 11'Ij;lbcnxv) = 1
is given by
(6.11) or to
Let PI (A) := sup (H>.J, f).
11/11=1
Thus, a lower bound of PleA) is given by
(6.12)
We point out that PleA) E a(H>.) because H>. is self-adjoint (see, for instance, [3] p. 96). We point out also that PleA) is not equal, a priori, to the norm of H>. since the latter may have, a priori, a negative spectrum if o is not convex or (in the case where 0 tJ. V) if A < - inf a(.). The estimate (6 .12) shows that PleA) > 0 and, consequently, PleA) is the greatest eigenvalue of the compact operator H>. . Thus
=
101- 1
~c
f
f f [f df-L(v)
n dx
S(x,w) 0
df-L(v) 1vfviVKcp(v) 12 =
e-
+00
,,(~) -,,(O) ) 1 12 I~I 8ds vfviJKcp(v)
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154
in view of (6.10) where
c = Inl
-. 1 [l I
mf
vEV
n
dx
S
(x,W)
e
-
,,(v)-,,(O) 8
Ivl
I
ds > O.
0
Since PI(>') depends continuously on >. > -0'(0) (see [2J Theorem 8, p. 213) and tends to zero at infinity, then there exists>. > -0'(0) such that PI (>.) = 1 and consequently (6.8) has a solution regardless of the size of n. This ends the proof. 0 We have seen, when K is bounded, that O'(T + K) n {>.; >. > -O'(O)} = 0 for n small enough. The following theorem gives more precise informations for constant collision frequencies. To this end we define some useful parameters. We note that dist(x, an)
Ivl
::; s
( )< x,v -
d
T0'
(6.13)
Let
s(v) :=
1~ll s(x, v)dx , 'Y:= 1~ll dist(x, an)dx.
It follows from (6.13) that 'Y
d
T0 ::; s(v) ::; T0 ' Let
K be the integral operator on L2(V; dJ..L) k(v,v')
(6.14)
with kernel
= Js(v)k(v,v')Js(v') .
It is clear, from (6.14), that
(6.15)
We have Theorem 6.4 Let K be bounded in L2(V; dJ..L) and let 0'( .) be constant. Then (i) O'(T + K) n {>.; >. > ->'*} = 0 if d 1.
155
Chapter 6. Form positive collision operators
Proof. (i) is a part of Theorem 6.3. Let cp E L2(V; dp,) with unit norm. Then the calculation in (6.11) shows that
=
101- 1
f f n dx
v dp,(v)
[f
S(x>V) 0
1
e-(C1+>')sds IVKcp(v)
I
2
from which we get
Hence, since cp E L2(V; dp,) is arbitrary with norm 1, taking the supremum yields lim >'~~C1P1('\) and we conclude as previously.
~ IIKII > 1
Remark 6.1 This theorem shows that the existence (or nonexistence) of eigenvalues is strongly related to the size of the geometrical parameters d and 'Y.
The following example gives more insight into the (probable) physical interpretation of the existence of eigenvalues Corollary 6.1 Let V be bounded and let k(v;v') = c be a constant. Then
O'(T + K) n {A;'\ > -O'} =I- 0 if
r
inxv
s(x,v)dxdp,(v)
> ~. c
Proof. Let cp E L2(V; dp,) be arbitrary with norm 1. Then
Thus
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156
so that
11K I = sUPII~II=lIIK\p1I = c
[Iv
= c
[1 v
s(v)dtt(v) 1= Inl
and we conclude by Theorem 6.4.
.i 2
s(V)dtt (v)1
I VsDIIL2
loxv s(x, v)dxdtt(v)
0
Remark 6.2 One sees that the existence of eigenvalues has to do with the magnitude of the parameter
r
ioxv
s(x, v)dxdtt(v)
which can be viewed as the "mean time", for the particles, to reach the boundary. Remark 6.3 If K is bounded then H).. has a limit, in the operator norm, as A ---+ -0"(0) H-a(O) :=
[JK ~1
O-u(O)
[~ JKj.
It follows that only .finitely many equations Pi(A) = 1 (i = 1,2 ...) may have solutions, since Pi(A) ---+ 0 as i ---+ 00 uniformly in A E [-0"(0),00[. It is to be expected that the curves Pi(')' which contribute to the spectrum ofT+K, give rise to only finitely many solutions. (This is true, of course, if the curves are decreasing.) In such a case, the point spectrum of T + K is .finite. This is certainly true if 1 is not an eigenvalue of H -u(O) ' Anyway the number of eigenvalues of H-u(o) exceeding 1 provides a lower bound of the number of eigenvalues of T + K. The point spectrum of T + K can be infinite as is shown in the following theorem. To this end we recall that the Radon transform of \P E Ll(V; dv) is given by R\p(w, s):=
l .v=s
\p(v)dv ; wE
Theorem 6.5 Let dtt(v) = dv and O"(v) exists q E L2(V) such that
= 0"
sn-\
s E R.
a constant. We assume there
lim R(IKqI2)(w, s) = +00.
8--+0
Then the point spectrum of O"(T + K) n {A > -O"} is infinite.
(6.16)
157
Chapter 6. Form positive collision operators
Proof. Let mEN be an arbitrary integer and Fm C L2(0) be a subspace of dimension m. Let cp E L2(V) with unit norm. We define also the mdimensional subspace of L2(0 x V)
According to the max-min principle (see [2] Theorem 5, p. 212),
where
1>. E Sm.
We note that
f>,(x, v) = 'l/J)..(x)cp(v) where
According to (6.9),
(H)..J>-.,
_ r dw 1'l/J)..(w) - 12 r (a(v .w)2 + >.) IJKcp(v) 12 + (a + >.)2 dv.
j)")L2(O XV) -
iRn
iv
We set h = IJKcp(v) 12 E £l(Rn) by extending it trivially outside V. Then
Pm(>')
~
/d
1;;;( w )..
2 )1 jd
W
.
Rn
R
8
/
(a+>.)h(v)
Iwl282+ (a + >.)2
v'I::;I=s
dw 1'l/J).. (w) 12 j d8-(a--;:.-->.)_R_h....!.(1:=1_'8_) .R Iwl 82+ (a + >.)2 Rn /
=
/
Rn
N ow we ch oose cp =
1
dw 'l/J)..(w)
VKql IIVKq I'
Th
12/ d8 Rh(I:I,(a+>.)p) 2 Iwl p2 + 1 R
en
h() IKq(v)12 v = IIVKql11 and
.
d V
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158
Letting Aj --t -a, the sequence {'l/J>.j; j E N} , included in the unit sphere of the finite dimensional space Fm, is relatively compact. We may assume that 'l/J>.j --t 'l/J in Fm (11'l/JIIL2(f2) = 1). Since ;;;;; --t
;j in L2(Rn)
then, in view of (6.16), Fatou's lemma yields
Hence, in view of the continuity of the curves Pi(.) ([2] Theorem 8, p. 213), there exist {31> {32, .. . , (3m such that P1.·({3t·) = 1
,
1_ < i_ <m
and this shows the existence of m eigenvalues {{3i; 1 :S i :S m} of T + K in view of (6.8). This ends the proof since m is arbitrary. Let us consider the case where 0 ~ V. Then the spectral bound of T
seT) = We recall that K : L2(V; dJ.L) with
--t
-00.
L2(V; dJ.L) is positive compact. We begin
Theorem 6.6 Let 1 :S M :S 00 be the number of positive eigenvalues of K. Then aCT + K) n R contains, at least, M eigenvalues. Proof Let E be the orthogonal (in L2(V)) subspace to the kernel of K. It is spanned by the eigenfunctions of K corresponding to the positive eigenvalues. Let m :S M (m finite), and Em C E be a subspace of dimension m. Let C c n be a ball. We define Oifx~C
'l/J(x) :=
{
ICI-~ if x
E C
where 101 is the Lebesgue measure of C. We define also the m-dimensional subspace of L2(n x V) 8 m :='l/J®Em. Let PI (A) 2: P2 (A) 2: .. . be the non-negative eigenvalues of the compact self-adjoint operator H>., each eigenvalue is repeated according to its multiplicity. According to the max-min principle (see [2] Theorem 5, p. 212),
159
Chapter 6. Form positive collision operators where f>.(x, v)
= 1P(x)u,X(v) ; U,X( .) E Em , Ilu,Xlb(V) = 1.
Hence
J [8(7)
> IOI-!
CxV
J
101- 1
e-('x+U(V))S1P(X - SV)dS] 1v'Ku,X (v) 12
0
[SjV) e-(,X+U(V))SdS]
CxV
lv'Ku,X(v) 12 > 0
(6.18)
0
where s(x, v) = inf {s > 0; x - sv rt. C} (x E C). By choosing a sequence Aj ---7 -00, {u,Xj; j E N} belongs to the unit sphere of the finite dimensional space Em. We may assume that u'xj
---7
U in Em ,
Ilull =
1.
Hence /Ku'xj ---7 /Ku in L2(V) and /Ku =I- 0 since u E Em C E. Finally, by Fatou's Lemma, lim inf Pm(A) = +00 >"j--+-OO
and we conclude as previously. 0 We end this section by showing a link between eigenvalues of T those of K .
+K
and
Theorem 6.7 Let Q1 2: Q2 2: ... 2: Qi·· · (i :s; M) be the eigenvalues of K. Let C c be a ball with radius r and let 0 < T < 1. We assume that V is bounded. Then any solution!3i of the equation Pi(!3) = 1 satisfies the estimate (l-T)r
n
T- n
r
Jo
6
e-({3,+u)8ds:S;
~ (1:S; i :s; M) Qi
where b is the maximum speed and ~ = max a(.) . Proof We use the arguments used in Theorem 6.6. We turn to (6.18) and choose, as m-dimensional subspace Em C E, the space spanned by the first m eigenfunctions of K . We have
Topics in Neutron llansport Theory
160 where Then
,(A)
UA
E Em. Let
2 CI-'
Zi be a ball concentric with C
/_cxv dxdp(v)
[$(x'v)
and with radius ~ r .
I
1
e-(A+u(v))sds~
1. 2
U( v ) A
It follows from
that
and this ends the proof.
6.4
0
Existence results for large spatial domains
+
This section is devoted to the behaviour of the point spectrum of T K when the spatial domain R gets large. Actually we are concerned with
u(T
+ K ) n {A > - inf a ( . ) ) .
We assume that R = kR1 where R1 c Rn is convex and k E R is intended to go to infinity. We observe that the spectral problem
is equivalent to
H(X,k)$ = a$
, $ E L2(R1 x V )
161
Chapter 6. Form positive collision operators
where 'Ij;(x, v) = ', k) is positive compact and that
2 r r (O"(v) + >.) 1v'K~(w, v)1 (H(>., k)'Ij;, 'Ij;) = dw dJ-L(v) "(v;.W)2 + (O"(v) + >.)2 JRn
(6.19)
Jv
Inl
'Ij;(x, v)e-ix.wdx. Let P1(>', k) 2: P2(>', k)··· where ~(w, v) = (27r)-n/2 be the eigenvalues of H(>', k) . We begin with the basic result Lemma 6.3 (i) Pm(>', k) < liSA II for all mEN, k E R, >. > - inf 0"(.). (ii) For all "X > - inf 0"(.) and mEN, Pm(>', k) --+ IISAII uniformly in
>. E ["X, oo[ as k --+
00.
Proof (i) It follows, from (6.19) and Parseval's identity, that \ k)·I'1',.•'I'1.) L2(nlXV} (H( A,
=
J J J J
=
IIMAv'K~1I2 ~ IISAIIII'Ij;11 2.
.,k)(x).,k)II£2(nd
= 1.
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162
Letting kj ~ 00, the sequence {-lP(>', kj);j E N} is relatively compact and we may assume that
'ljJ(>., kj ) ~ 'ljJ(>.) E Fm , Hence (i;(>., kJ· )
~
(i;(>.) in L2(Rn) and
11'ljJ(>.) II = 1.
II(i;(>.) I
1.
L2(Rn)
By Fatou's
lemma we deduce, from (6.20), lim inf Pm(>' , kj) 2: k;--+oo
2
ivrdJL(v) 1 J a(v )+ >. 1 JKcp(v)
Since cp is arbitrary,
limki~ooPm(>.,kj) 2: IIM"JKI1
2
=
J
IIS"II·
Hence, in view of (i),
and finally It is not difficult to see that the convergence is uniform in >. E [X, 00 [. We are in a position to prove the main result of this section. We define the size of 0 as the radius of the greatest ball included in O. We have
Theorem 6.8 a(T + K) n {>. > - inf a(.)} i:- 0 for 0 large enough if and only if a(B) n {A; >. > - inf a(.)} i:- 0. In such a case, the number of eigenvalues of T + K increases indefinitely with the size of 0 and all these eigenvalues converge to X the leading eigenvalue of B . Proof There is no loss of generality in assuming that 0 = kO l where 0 1 is fixed. According to the observations before Lemma 6.3, >. > - inf a(.) is an eigenvalue of T + K if and only if 1 is an eigenvalue of H(>', k) in L2(01 X V) . According to Lemma 6.1,
a(B) n {A; >.
> - inf a(.)}
=
0
if IIS"II < 1 for all >. > - inf a( .). In such a case, according to Lemma 6.3, IIH()., k)1I < 1 for all k E R and all >. > - inf a(.) so that a(T +
163
Chapter 6. Form positive collision operators
K) n {>. > - inf a(.)} is empty regardless of the size of n. Conversely, if a(B) n {>.; >. > - inf a(.)} =f. 0 then, according to Lemma 6.1, >. > -infa( .) -->
IIS,\II
is strictly decreasing
IISrII
and B has a leading eigenvalue 'X defined by = 1. Let /3 < 'X be arbitrary and let mEN be arbitrary also. Then, according to Lemma 6.3, Pm(>. , k) --> IIS,\II as k --> 00 uniformly in [/3,'X]. Therefore
PI (/3, k)
~
P2(/3, k)
~
...
~
Pm (/3, k) > 1 for k large enough
and finally there exist /31, /32, ... , /3m E ] /3, 'X [ such that Pi (/3i, k) = 1 , i = 1, 2, ... , m. This ends the proof. Remark 6.4 We already observed, in a more general setting, that the point spectrum of T + K is empty regardless of the size of n if the point spectrum of B is empty and that the converse result is not true (see Sections 5.5 and 5.6). Theorem 6.8 above shows that the converse is true for the model of this chapter. We note that the evenness assumptions (6.1) (6.2) exclude the counterexample pointed out in Remark 5.1.
6.5
The isotropic models
This section is devoted to additional spectral results pertaining to the assumption that the cross sections are isotropic, that is independent of the directions of velocities. To this end we assume that
dJ..L(v) := da(p)
@
dr(w) , P =
lvi,
v w= P
where da(p) is a Radon measure on [a , b] (0 ~ a < b ~ 00), and dr(w) is the surface Lebesgue measure on the unit sphere of Rn . Here,
Acp
8cp -a(p)cp(x,p,w) + = -PW'-8 x
lb
I
I
k(p,p )da(p)
a
Is
and
is self-adjoint compact. We observe that the spectral problem
Acp = >.cp; Re>. > s(T)
I
I
I
cp(x,p ,w )dr(w)
Sn-l
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164 amounts to 1
cp(x, p, w} = -
p
l
S(x,W)
e
('>'+u(p»' P
.
K1/J(x - sw, p}ds
(6.21)
0
where
hn-,
1/J(x,p} =
cp(x, p, W'}dT(W').
Further, for convex 0, (6.21) is equivalent to _ (.>.+,,(p»
1/J(x,p} =
1 n
dx
,e
1",-/1
p
pix - x
,
2
,
n-1
I
K1/J(x ,p}; 1/J E L (0 x [a,b]) .
(6.22)
The main feature of isotropic models is the following Theorem 6.9 Let the collision operator be self-adjoint and isotropic. Then the point spectrum of A in the half plane {ReA> - inf a( .}} is real. Proof. Suppose there exists a nonreal eigenvalue A = A1
+ iA2
(A2
¥
O). We define
E A : cp E L2 (0 x [a,b])
-t
1
e
1 1
(.>.+u(p»
x-x'
p
n pix - x
, n-1
I
cp(x " ,p}dx .
The spectral problem (6.22), i.e.
implies, in view of the self-adjointness of K, that
(E AK1/J,K1/J) is real (K1/J
¥ O) .
On the other hand, using Fourier transform with respect to the space variable and Parseval's identity,
where
165
Chapter 6. Form positive collision operators and
K;fi(w,p) =
(27r~n/2l K7/J(x,p)e- iw .x dx .
Hence
and therefore
because
Im(~(p, w)) =0
(6.23)
IK7/J12 (w , p) > O. On the other hand, 00
~(p, w) = ~ / e_ c>.+upCp»r dr / o
1/
Sn-l
e- ~ p dr
o
Let us set
p(s) =
/ 1
00
p
e-irw'. wdr(w' )
e-,·rw8 11 ds
-1
L,.f,j;r
=8
/ WI .
dr(w'); s
E
1:1=5
[-1 , 1]
{w'
the (n - 2)-dimensional measure of the set E sn-l; note that p(±I) = 0 and that p(.) is even. Then
w' . 1:1 = s} . We
~(p,w) 1
p(s) ds / A+O"(p)+iplwls
-1 1
/ [A + O"(p/+ ip IWI s o 1
2/3J o
+ >. + O"(p/- ip Iwl s Jp(s)ds 00
p(S) ds=_2_J p(I/t) /3 dt /32 + p21wl 2 S2 p IWI (-.1!1..)2 + 1 p Iwl (6.24) 1
plwl
where /3 = A + O"(p) . Let /3 = /31 + ifh. Using the fact that /31 > 0 and integrating by parts, one sees that the imaginary part of ~(p, w) is equal
Topics in Neutron Transport Theory
166
11
to
[(1- ~)2 + (~)2l
00
-2plwll
I
p(l/t)Log (1+~)2+(~)2
dt t2 '
Finally, since pi (lit) < 0,
~ ) { < 0 for Im'\ > 0 1m E>.(p, w) > 0 for Im.\ < 0
(
and this contradicts (6.23). For the rest of this section, we assume that K is positive in the scalar product sense. Then.\ is an eigenvalue of A if and only if 1 is an eigenvalue of the self-adjoint compact operator in L2(n x [a, b])
H>.
:=
.JKE>..JK.
On the other hand,
(H>.1f;,1f;) =
lb kn da(p)
E;,(p,w)
1.JK1f;1
2
(w,p)dw
and, by (6.24), for real.\
E;,(p, w) = 2(.\ + O'(p))
r p(s) Jo (.\ + a(p))2 + p21wl 1
2
s2
ds > O.
(6.25)
Thus H>. is positive. We define the operator
- lb K :
. > -O'(O)} consists of m real eigenvalues where m is the number of eigenvalues of H -') ;::: 1I2(>')'" be the eigenvalues of H>.. Arguing as in Remark 6.3, it suffices to prove that the curves
>. - t lIi(>')
i = 1,2, ...
are strictly decreasing. To this end it suffices that H>. - H{3 (>. < (3) be a positive operator in the scalar product sense. This is true if
>. - t E>.(p,w) is stricly decreasing. By integrating by parts the last integral in (6.24), E>.(p,w)
2
= plwl
[{OO I
p (l/t)tan-
il
(>.+O'(p))t
1
plwl
(
)
dt t2
7r
+ '2 P(O)
1
which ends the proof. 0 We complement Theorem 6.10 with a converse result. Theorem 6.11 We assume that K is not bounded in L2([0, b]). O'(A) n {Re>. > -O'(O)} consists of infinitely many real eigenvalues.
Then
Proof It suffices to prove that
lim
>. ..... -. remains selfadjoint. However, the evenness assmnption (6.2) with respect to each variable is crucial. It would be very useful, but not so easy, to extend the theory above by replacing the strong evenness assmnption (6.2) by the more physicalone k(-v,-v') = k(v,v'). We point out that Theorem 6.4 admits a straightforward version for nonconstant collision frequencies. For isotropic models (see Theorem 6.10), the nmnber of eigenvalues of the transport operator is equal to the nmnber of eigenvalues exceeding one of the compact self-adjoint operator H - 0 and j :2: 0, {Uj(t) - Uj(t)j 0::; t ::; to} is collectively compact. Proof: We recall that M C L(X) is said to be collectively compact if there exists a relatively compact subset of X containing A(Bx) for all A E M where B x is the unit ball of X . We prove the lemma by induction on j :2: 0 assuming it is true for j - 1. We note that, for x E D(T), Uj(t)x - Uj(t)x
=
=
J: It
.
+
[Uj-l(t - s)B - Uj_l(t - s)B] U(s)xds
Uj_l(t - s)(B - B)U(s)xds
0
J:
[Ui-l(t - s) - Uj-l(t - s)] BU(s)xds.
Let c = sup {I/U(s)11 j 0::; s ::; to}. Then (B - B)U(s)(Bx)
c
K(cBx)
j
0::; s::; to.
The strong continuity of Ui-l(.) implies C = {Ui-l(S)K(cBx)jO::; s::; to} is relatively compact
(7.13)
O}
181
Chapter 7. On Miyadera perturbations of co-semigroups
and
lt
Uj-1(t - s)(.8 - B)U(s)ds(Bx) C toconv(C) ; t::; to.
(7.14)
On the other hand, by the Miyadera condition, there exists "(' 2: 0 such that
l
ta
IIBU(s)xll ds ::; "('
Ilxll ;
x E D(T).
Let
d = {(Uj_1(s) -
Uj_1(s)) (Bx);O::; s::; to}
which is relatively compact by the induction hypothesis. Let x E D(T), Ilxll ::; 1 and 0 ::; t ::; to. If x' E X' is such that
:~g, 1( x' , y) 1 ::; 1 (i.e. x' E (C')O the polar set of C' ) then
1( x' ,.f: [Uj-l(t -
: ; J: I(
s) - Uj-l(t - s)] BU(s)xds) 1
x' , [Uj_1(t - s) - Uj_l(t - s)] BU(s)x)1 ds
::; J>BU(S)X II ds ::; "(' . According to the bipolar theorem ([8] Theorem 2, p. 137), we deduce
lt
[Uj_1(t - s) - Uj_1(t - s)] BU(s)xds
E "(' conv(C ' ).
(7.15)
Finally, by combining (7 .14) and (7.15), we obtain
(Uj(t) - Uj(t)) (Bx) C toconv(C) and then {(Uj(t) - Uj(t)) (Bx);O::;
+ "(' conv(C' ) ,0::; t ::; to
t::; to}
is relatively compact.
Lemma 7.3 Let w' > we(U) and n E N. Then there exists a constant en(w ' ) such that n
re(LUj(t)) ::; cn(w')etw' j=O
(t 2: 0) .
Topics in Neutron '1Tansport Theory
182
Proof Let PI be the projection corresponding to O"(T) n {A; ReA :2: w' } . We note that PI is a finite rank projection commuting with {U(t); t :2: O} and the type of {U(t)IXe; t :2: O} is < w' where Xe = Pe(X) and Pe = I - PI . In particular, there exists M:2: 1 such that
The subspaces XI = PI(X) ; Xe = Pe(X) are invariant under {U(t); t :2: O} and XI C D(T) (dimXI < 00). It follows that
Pe E L(D(T); D(T)) . Let
-
-
13 = PeBPe. Then 13 E L(D(T);X)
.I:
IIBU(t)xll dt
J: :.:; J:
=
and, for x E D(T),
IIPeBPeU(t)xll dt
=
J:
IIPeBU(t)Pexll dt
IIBU(t)Pexll dt :.::; 'Y IIPex11
:.: ; 'Y Ilxll·
Hence 13 is a Miyadera perturbation of T . We denote by Uj(t) (j:2: 0) the corresponding iterations. It follows from
that Uj(t) leaves Xe invariant and vanishes on XI' Moreover, the parts of
Uj(t) on Xe are nothing but the iterations corresponding to U(')IXe and the Miyadera perturbation PeBIXenD(T)' According to Lemma 7.1, there exists a constant en (w') such that
Since (Xe , XI) is a pair of reducing subspaces for 00, it follows that
r.
[t,
U;
2:7=0 Uj(t) and dim XI
(t)]-; c,,(w' )e"'" .
we(U) . 0
184
7.4
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Comments
The generation theorem for Miyadera perturbations was given first (of course!) by 1. Miyadera [2] and is stated there in a different form. The statement we give is taken from J. Voigt [5] where it is shown, in particular, that the perturbed semigroup is given by a Dyson-Phillips expansion (as for bounded perturbations) for small times. The convergence of the Dyson-Phillips expansion for all times as well as the estimates of the terms ofthe series (Lemma 7.1) are due to A. Rhandi [4] who also proved that the essential type of the perturbed semigroup is less than or equal to the type of the unperturbed semigroup. The inequality between the essential types (Theorem 7.2) is due to J . Voigt [7] and actually holds under more general assumptions. We point out that the stability of the essential type for (unbounded) Miyadera perturbations is an open problem; the difficulty being that B (or-B) is not (apparently) a Miyadera perturbation of A = T + B. Finally, a useful open problem is to give sufficient conditions (in terms of the unperturbed semigroup and the perturbation B) implying the assumptions of Theorem 7.2.
References [1] W . Arendt and A. Rhandi. Perturbations of positive semigroups. Arch. Math. 56 (1991) 107-119. [2] 1. Miyadera. On perturbation theory for semigroups of operators. Tohoku. Math. J. 18 (1966) 299-310. [3] M. Mokhtar-Kharroubi. Characterisation BV des perturbations de Miyadera et applications. Workshop "Evolution equations, Control theory, Biomathematics". Luminy, March 8-12, 1993. [4] A. Rhandi. Perturbations positives des equations d'evolution et applications. These, Universite de F'ranche-Comte Besanc:;on, (1990). [5] J. Voigt. On the perturbation theory for strongly continuous semigroups. Math. Ann. 229 (1977) 163-17l. [6] J. Voigt. On substochastic CO-semigroups and their generators. Semes terbericht Funktionalanalysis, Thbingen, Wintersemester, (1984/85). [7] J. Voigt. Stability of the essential type of strongly continuous semigroups. Trans. Steklov. Math Inst. 203 (1994) 469-477. [8] K. Yosida. Functional Analysis. Springer Verlag, 1978.
Chapter 8
On resolvent positive operators and positive co-semigroups in Ll (J-L) spaces 8.1
Introduction
The aim of this chapter is to present some mathematical results about Desch's perturbation theorem [5] . This theorem emphasizes the particular role of positivity in perturbation theory in L1 spaces. Besides its mathematical interest, this perturbation theorem proves useful for the treatment of transport equations with singular cross-sections (i.e. unbounded collision frequencies and unbounded collision operators) which are considered in Chapter 9. This remarkable role of positivity, tied to the additivity of the L1 norm on the positive cone, appears also in the context of scattering theory in Chapter 12 and in the mathematical analysis of singular transport equations in Chapter 9. Let us state Desch's result [5] . TheoreIn 8.1 Let {U(t)j t :?: O} be a positive co-semigroups in L1(p,) with generator T and let B E L(D(T) j £1 (p,)) be a positive operator. Let there exist>. > s(T) (the spectral bound ofT) such that (>.-T-B)-l exists and is positive. Then T + B generates a (positive) co-semigroup.
We point out that this result is not true in V(p,) spaces (1 < p < 00) [2]. We do not present Desch's proof but rather different proofs which are 185
186
Topics in Neutron Transport Theory
somewhat simpler than the original one and provide useful informations on the perturbed semigroup in preparation for the spectral analysis of singular transport equations we deal with in Chapter 9. The first proof we give relies on repeated applications of the Miyadera perturbation theorem described in Chapter 7. The second one, based directly on Miyadera perturbation theorem, relies on renorming arguments. Finally, we provide a third alternative proof independent of the Miyadera perturbation theorem.
8.2
A preliminary result
Let X be a Banach lattice and T be an unbounded linear operator acting on X with domain D(T). We recall that the spectral bound of T is defined as s(T) = sup {ReA; A E a(T)} . We say that T is resolvent positive if s(T) < for A > s(T). We start with
00
and (A - T)-l is positive
Lemma 8.1 Let X be a Banach lattice, T be a resolvent positive operator and A > s(T) . Let B E L(D(T); L1(J.L)) be a positive operator. Then the following conditions are equivalent. (i) r 17 (B(A - T)-l) < 1. (ii) A E p(T + B) and (A - T - B)-l is positive.
Proof: Let A > s(T) be such that r 17 (B(A - T)-l) problem AX - T x - Bx = y ; X E D(T).
. - T - B)-l = (A - T)-l ~)B(A - T)-l)n ~ (A - T)-l ~ o. n=O Conversely, let A > s(T) be such that (A-T-B)-l ~ O. Then the identity n
(A - T - B}2:(A - T)-l(B(A - T)-l)j = I - (B(A - T)-l)n+l j=O
implies
n+1 2:(B(A-T)-l)j = B(A-T-B)-l(I _(B(A_T)-1)n+1) ::; B(A-T-B)-l j=l whence rq(B(A - T)-l) ::; 1 and, for any J.L > 1, [J.L - B(A - T)-l] -1
=
f
(B(A ;~)-1 )j ::; I
+ B(A -
T _ B)-l
j=O
so that
On the other hand (see, for instance, [4] Proposition 2.19, p. 96),
because the spectral radius of a positive operator belongs to its spectrum (Theorem 5.1) and therefore (8.2) shows that rq(B(A - T)-l) < 1. 0
8.3
Miyadera perturbations in L 1 (J.L) spaces
We start with a weaker version of Desch's result. Lemma 8.2 Let T be the generator of a positive co-semigroup in L 1(J.L) and let BE L(D(T); L 1(J.L)) be a positive operator. We assume there exists A > s(T) such that IIB(A - T)-lll < 1. Then T + B generates a (positive) co-semigroup.
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188
Proof: Let x E D(T), x ~ O. Then, because of the additivity of the U norm on the positive cone,
J~ IIBe-AtU(t)x ll dt
= liB
J~ e-AtU(t)xdt ll
= IIB(A -
T)-lxll
::; IIB(A - T) -lllllxli whence, for all a:
> 0,
lD.IIBe-AtU(t)XII dt::; 'Y Ilxll
; x E D(T)
, x
~0
where 1
1
Let x E D(T) and x~ = n Jon U(t)x+dt , x:: = n Jon U(t)x- dt where x+ (resp. x-) is the positive part (resp. the negative part) of x. It is easy to see that
x~
-
x~
-->
x in D(T) as n
--> 00.
(8 .3)
From
we pass to the limit , thanks to (8 .3) , and get
i.e. B is a Miyadera perturbation of the generator T - AI. The generation result follows from the results of Chapter 7. We are now ready to prove
Theorem 8.2 Let T be the generator of a positive eo-semigroup in L 1 (J1.) and let BE L(D(T);U(J1.)) be a positive operator. We assume there exists A > s(T) such that (A - T - B)-1 is positive. Then T + B generates a (positive) eo-semigroup.
Chapter 8. On positive co-semigroups in Ll (J.L) spaces
189
Proof: According to Lemma 8.1, ru(B().. - T)-l) < 1. By replacing B by sB, s E [0,1]' Lemma 8.1 asserts that
Let n E N be such that
Then
In particular,
Thus, (8.4) allows us to apply Lemma 8.2 repeatedly for j = 0, 1, ... , n-l with the perturbation n- l B and this ends the proof. Because of the repeated use of the Miyadera perturbation theorem it is not clear, a priori, that the perturbed semigroup is given by a DysonPhillips expansion. To prove it is so we need the following result. Lemma 8.3 Let C E L(Ll(J.L)) be a positive operator such that ru(C) < 1. We denote by II the usual Ll norm. Then there exists an equivalent norm IlIon Ll(J.L) which is additive on the positive cone and such that the corresponding operator norm of C is less than one.
Proof: Let c and c' be two constants such that no E N be such that
°< c < c' < 1 and let
We set
(8.5) It is easy to see that 1.11 is a norm in Ll(J.L) and is additive on the positive cone. Moreover, it is clear that
190
Topics in Neutron Transport Theory
and
::::;
o ,,00 (~C)OIl)k "n ~p=O (c'Y ~k=O c' no Ixl
::::;
o .lJQ!]. ,,00 ( no ) "n ~p=O (c'Y ~k=O "Pro
-1.lJQ!].
-1
k
Ixl
which shows the equivalence of the two norms. Finally
whence
IIGll 1 ::::; c' < 1
where 11111 is the operator norm associated with 111'
0
Theorem 8.3 Under the assumtions of Theorem 8.2 the semigroup generated by T + B is given by the Dyson-Phillips expansion. Proof: We denote by II the usual P norm. According to Lemma 8.1, there exists). > s(T) such that r 17 (B(). - T)-l) < 1. In view of Lemma 8.3, there exists a norm 111 equivalent to II, additive on the positive cone and such that IIB(). - T)-1111 < 1. By using the new norm, the proof of Lemma 8.2 shows that B is a Miyadera perturbation of T - ),1 and, according to the general theory of Chapter 7, e-Atet(T+B) =
00 L: Uj(t)
(8.6)
j=O
where Uj+l(t) = lot Uj(t - s)Be-ASU(s)ds , Uo(t) = e-AtU(t) . The series (8.6) converges in the operator norm 11111 (and therefore in the usual operator norm) uniformly in bounded times. On the other hand, the convergence of the series (8.6) is obviously equivalent to
00
et(T+B) = L:Uj(t) j=l
Chapter 8. On positive co-semigroups in L 1 (JL) spaces
191
where
Uj+1(t) = lot Uj(t - s)BU(s)ds
j
Uo(t) = U(t)
which ends the proof. c). Therefore it suffices to give a proof for a class of operators dense in L(L~(Rn); L1(Rn)). By approximating K in L(L~(Rn); L1(Rn)) by finite rank operators (and using the linearity) it suffices to prove that
IIK().. - T)-l H().. -
T)-ll1
--+
0 as Re)..
--+ 00
where K and H are one rank operators with kernels
K(v,v')
= f(v)g(v')
, H(v,v')
= h(v)k(v')
where f, hE L1(Rn) and ;, ~ E Loo(Rn) . In view of Proposition 9.2, we have to prove that IIK().. - T)-l HIIL(L~(nXRn);£l(nXRn))
--+
0 as Re)..
--+ 00 .
We factorize K()" - T)-l Has
K().. - T)-l H = F.N>. .k where
k : 'l/J F : 'l/J
E
E
N>.: 'l/J
L~(n x L1(n) E
Rn)
--+
J
'l/J(x, v)k(v)dv
E
L1(n)
Rn
--+
L1(n)
'l/J(x)f(v)
--+
J
E
L1(n x Rn)
n N>.(x,y)'l/J(y)dy E L1(n)
Chapter 9. On singular transport equations in L1 spaces
201
and
rOO
N>.(x , y) = Jo
(>.
e- +"
(=» t
x - y x- y x - y dt tg(-t-)h(-t-)x(lx-yl::::s(x ' lx_yl))t n
It suffices that
(9.10) It is easy to see that ( Ig(z)llh(z)1 IIN>.IIL(Ll(n»:::: JRn Re,X+O'(z)dz
(9.11)
and there exists c, independent of hand Re'x ~ c' > c, such that (9.12) The estimate (9.12) allows us to assume, without loss of generality, that h is compactly supported and its support is included in R!' - A where A is the set of singularities of the collision frequency 0'( .). The estimate (9.11) becomes Ig(z)llh(z)1 (9.13) IIN>.IIL(L'(n» :::: Re'x + O'(Z) dz
1
supp(h)
and the boundedness of 0'( .) on supp(h) implies (9.10) . . --+ 00. For instance, in the whole space, i.e. = R n , and for
n
>'+c>O
IIK(>. - T)-111 = sup >. vERn
where k(v)
= J K(v', v)dv' . By
assumption
k(V~ ) + 0' V
*1 is bounded. If *1 ~ a > 0
in an open subset of Rn on which 0'( .) is not bounded, then SUPvERn >.~~iv) ~ a regardless of >.. Proof of Theorem 9.2: In view of Proposition 9.3, r,,(K(>' - T)-l) < 1 for>. large enough. It follows, from Lemma 8.1, that (>. - T - K)-l exists and is positive for >. large enough and we conclude by Theorem 8.2. s(T)} consists of, at most, isolated eigenvalues with finite algebraic multiplicities. Since this is not sufficient, a priori, to infer the spectral properties of the perturbed semigroup, a direct analysis is considered in the following section.
9.3
The essential type of the perturbed semigroup
According to the theory in Chapter 7 (Theorem 7.2), the essential type of {V(t); t ~ O} is less than or equal to that of {U(t); t ~ O} provided that some remainder term of the Dyson-Phillips expansion is weakly compact in £l(n x Rn). We will deal with the second order term R2(t) . Two major difficulties arise. First, in the theory of Chapter 7, the terms of the Dyson-Phillips expansion are not explicit on the whole space. Their definition on the whole space is obtained by an (abstract) extension argument. Secondly, those terms involve strong time integrals of operators which are not locally bounded in time. It is to overcome such difficulties that several technical preliminary results are needed. First of all, the smoothing effect of {U(t); t ~ O} (Proposition 9.1) is inherited by V(t) = et(T+K) in the following form. Proposition 9.4 For any a > 0 there exists a constant M such that
The proof is based on another technical result we prove first. Let a > 0 be fixed and let AO be large enough so that (9.15) Let E be the space of strongly measurable mappings
i.e. for any f E £l(n x Rn) t E [0, a[
-+
Z(t)f E L1(n x Rn) is measurable
204
Topics in Neutron Transport Theory
and such that
endowed with the norm
IIZIIE =
sup 11/11:9
r IIZ(s)fll ds.
(9.16)
10
The finiteness of IIZIIE follows from Baire's theorem (see, for instance, [IJ p. 15). We define the operator
L: Z
E
E
---7
LZ(t)
=
1t
Z(s)KU(t - s)ds ; t
E
[O,aJ
where U(t) = e->'otU(t). Then Lemma 9.1 L E L(E) and Ta(L) ::; Ta(K(AO - T)-1) .
Proof: Let f
E
L1(n x Rn) be non-negative, then LZ(t)f
=
1t
Z(s)KU(t - s)fds.
Hence, thanks to the additivity L1 norm on the positive cone,
J:
IILZ(t)fll dt
: ; J: J: : ; J: J~ : ; J~ dt
ds
J: J: = J~ J:
IIZ(s)KU(t - s)fll ds = IIZ(s)KU(T)fll dT
dT IIZIIE IIKU(T)fll =
ds
dT
IIZ(s)KU(t - s)fll dt
IIZ(s)KU(T)fll ds
IIZIIE J~ IIKe->'oTU(T)fll dT
= IIZIIE IIJ~ Ke->,oTU(T)fdTII = IIZIIE IIK(Ao - T)-1 f II·
Chapter 9. On singular transport equations in Ll spaces If 1 E Ll(n x R"') is arbitrary then the above calculation shows that
J:
IILZ(t)III dt
:::;
IIZIIE [lIK(>.o -
T)-l
1+11 + IIK(>.o - T)-l 1-11]
:::; IIZIIE IIK(>.o - T)-lll [111+11 + 111-111 =
IIZII EIIK('\o - T)-lIIIlIII ·
Thus
and
On the other hand, Lm Z(t)I is equal to
and, for non-negative
J:
1,
IILm Z(t)III dt
Making the change of variables
tm - t m - 1 = U m t -tm = Um+l
205
206
Topics in Neutron Transport Theory
and using Fubini's theorem we get
I:
IILm Z(t)/11 dt
: ; I: IUi~O dUI
=
.IUi~O
IIZ(Ul)KU(U2) ·· · KU(u m+1)/11 dU2· · · du mdu m+1
dU2 ··· dumdum+1
I:
dU11IZ(UI)KU(U2)· ·· KU(u m+1)/11
:::; IIZIIE.IUi~O dU2 ·· · dumdum+l IIKU(U2) ··· KU(um+1)/11 = IIZIIE
IIIUi~O
dU2 ··· dU mdU m+1 KU (U2) ·· · KU(um+d/ll
= IIZII E I [K(>.o - T)-I]m III· When IE Ll([2
I:
X
Rn) is arbitrary we obtain, by decomposing it,
IILm Z(t)/11 dt
:::; IIZII E [II [K(>.o - T)-I]m 1+11 + I [K(>.o - T)-I]m I-II] :::; IIZII EI [K(>.o - T)-I]mll [111+11 + III-Ill =
IIZIIE I [K(>.o - T)-I]mllll/ll ·
Hence
and
Thus (9.17)
which proves the claim.
207
Chapter 9. On singular transport equations in L1 spaces
Proof of Proposition 9.4 : We observe that
=
Ja J o
le-t(>'o+a(v)) f(x - tv, v)X(t < s(x, v))1 a(v)dxdv
dt
OxRn
:; 1 1 00
le-t(>'o+a(v» f(y, v)1 a(v)dydv ,
dt
.
0
whence
.
OxRn
.otV(t). Then Z(t) = aV(t) satisfies the Fredholm equation
Z
= aU +LZ.
One sees, in view of (9.15) and (9.17), that
Z = (I - L)-lU E E which proves our claim since V(t) = e>.otV(t). Thanks to the smoothing effects of {V(t); t 2 O} one easily derives the following results whose proofs are left to the reader Proposition 9.5 The following Duhamel equation holds
V(t)f
= U(t)f + lot U(t - s)KV(s)fds ; f
E L1(0
X
Rn)
(9.20)
and the second order term of the Dyson-Phillips expansion is given by
Topics in Neutron Transport Theory
208
For the sequel we need the following stability result. Proposition 9.6 There exists c(t) locally bounded in t , independent of K,
such that
Proof. According to (9.21) , t
II R2(t)fll
::; Cl(t)
JJ 8
ds
o
0
JJ t
=
Cl(t)
t
dr
o
ds IIKU(s - r)KV(r)fll
r
J t
< Cl(t)
dr IIKU(s - r)KV(r)fll
t
dr / dp IIKU(p)KV(r)fll
o
(9.23)
0
where Cl(t) = sUPo(supp(h» II~"L'''> IlcpliL~ which proves (9.26) . Thanks to (9.26) , we have
Illgfll ::; M
I: ds I:-g dr
IIV(r)fIIL~
= M It It X(E , t)(s)X(s - c:, s)(r) IIV(r)fll£1 drds o
whence
0
u
Illgli ::; c:MsuPllfIlL,9 J~ IIV(r)fIIL~ dr.
We point out that sup
{t IiV(r)fllL' dr < 00
IIfllLl9 Jo
u
,
)dydv
,
211
Chapter 9. On singular transport equations in L1 spaces
in view of Baire's theorem (see (9.16) and the proof of Proposition 9.4). Finally 111,,11 - t 0 as c - t o. We prove, by similar arguments, that 111,,11 - t 0 as c - t O. We are now ready to prove the main result of this section.
Theorem 9.3 We assume that the collision operator
is dominated by a compact operator and that
n is bounded.
Then R2(t) is
weakly compact in L1(n x Rn) . Proof We first assume that
is compact. In view of the preliminary results, it suffices that m(t) be weakly compact in L1(n x Rn). Let 'I/J E L~(n x Rn), then
U(t - s)KU(s - r)H'I/J =
r
k(x, v, r, s, x', V')'I/J(X/, v')dx' dv '
inxRn
where k(x,v,r,s,x/,v ' ) is equal to
m(t - s,x,v) h(V)gl(X - x' - (t - s)v) (s-r)n s-r xm ( s-r,x-(t-s)v, and m(t,x,v)
X -
x' - (t - s)V) (X - x' - (t - s)V) I f2 g2(V) s-r s-r
= e-tcr(v)X(t < s(x,v».
We factorize m(t) as
m(t) = Q2Q1 with
Q1. , so that r a (K (>. - T) -1) < 1 for >. large enough because of the upper semicontinuity (with respect to the operator norm) of the ~pectral .radius ([3] p. 20~). W~ conjecture that IIK21IL(L~(Rn);L'(Rn)) < 1 IS suffic2ent. The analysIs of smgular neutron transport equations in £P spaces (1 < p < 00) is dealt with by M. Mokhtar-Kharroubi and J. Voigt
[4].
214
Topics in Neutron Transport Theory
References [1] H. Brezis. Analyse Fonctionnelle: Theorie et Applications. Masson, Paris, 1983. [2] M. Chabi. TMorie de scattering dans les espaces de Banach reticules. Transport singulier dans £1. These de l'Universite de Franche-Comte, 1995. [3] T . Kato. Perturbation Theory for Linear Operators. Springer Verlag, 1984. [4] M. Mokhtar-Kharroubi and J. Voigt. On singular neutron transport equations in V spaces. Work in preparation. [5] A. Suhadolc. Linearized Boltzmann equation in L1 space. J. Math. Anal. Appl. 35 (1971) 1-13. [6] J . Voigt. On substochastic eo-semigroups and their generators. Semesterbericht Funktionalanalysis, Thbingen, Wintersemester, (1984/85), 115.
Chapter 10
Stochastic formulations of neutron transport. Nonlinear problems 10.1
Introduction
This chapter deals with a class of nonlinear neutron transport problems arising in the probabilistic approach of neutron chain fissions . Neutron transport theory deals ordinarily with expected value of neutron populations . In order to describe the fluctuations from the mean value of neutron distributions , stochastic formulations of neutron chain fissions have been introduced very early, in particular by L. Pal [11], G.I. Bell [3] [1] and M. Otsuka and K. Saito [10] . Two main mathematical problems, related to such formulations, are considered in this chapter. We first study the stationary equation
8r.p v.--- -a(x, v)r.p(x, v) 8x -a(x, v) x
[1 - co(x, v) - f
J
Ck( X,
v,
v~ , ... , v~)
k=1 Vk , " x (l- r.p ( x , VI) · ·· (1- r.p(x, vk)dv i
. ..
I dVk))
1
(10.1)
with the conditions (10.2) 215
216 where m
Topics in Neutron Transport Theory ~
En x V and r + = {(x, v) E an x V; v.n(x) > O}.
2 is an integer, (x, v)
We use the notation Vk = V X . . . X V (k times) and assume that the velocity space V is the unit ball of R':; with normalized Lebesgue measure. Next, we will deal with the evolution problem
a'l/J at - v. a'l/J ax + a(x, v)'l/J(t, x, v)
a(x,v) x [l-CO(X,V) -
f. JCk(X,V,V~,
...
,v~)
k=l Vk
x(l-
'l/J(t,x,v~) ... (1- 'l/J(t,x,v~)dv~ ... dV~))l
(10.3)
with the conditions
'l/J(O, x , v) = 'l/Jo(x, v) , 'l/J(t,., .)Ir + = 0 , 0::; 'l/J ::; 1.
(lOA)
We also study the time asymptotic behaviour of its solutions. Let us recall briefly the physical meaning of the equations above and refer to G.I. Bell [3] [1] for details. In a multiplying medium occupying a region ncR';, a neutron, interacting with the nucleus of the host material, may be absorbed, scattered in random directions or may produce (instantaneously), by fission process, more than one neutron. The probability that a neutron, with velocity v E V, at position x E n, yields, by fission process, i neutrons (1 ::; i ::; m) with velocities v~, ... , v: is denoted by
and
(10.5) where eo(x, v) is the probability to be absorbed. An important information is provided by the probability pj(tf, x, v, t) j
= 0, I, ...
that a neutron, born at time t with velocity v and position x, gives rise to > t. The probabilities Pj(t f, x, v, t) (j = 0, 1, ... ) are
j neutrons at time t f
217
Chapter 10. Nonlinear problems
governed by infinitely many coupled equations [15] [16]. On the other hand, the probability generating function 00
G(z,x,v,t,tf):= L,Zipi(tf,X,v,t); t
< tf
o is governed by a nonlinear backward equation (see [3]) with the final condition G(z,x,v,tf,tf) = z and the boundary condition G(z,x,v,t,tf) = 1; (x,v) E r+, t
< tf .
This chapter is devoted to the mathematical analysis of the probability generating function. Mathematically speaking it is expedient to consider 7/J(z, x , v, t) := 1 - G(z, x , v, tf - t, tf)
which is governed by the initial boundary value problem (10.3) with initial condition 7/Jo(x, v) = 1 - z and homogeneous boundary condition. Equation (10.1) governs the probability of a divergent chain reaction. We note, in view of (10.5), that the stationary problem (10.1) (10.2) admits
.* > O.
(10.6)
We formulate (10.1) (10.2) as a fixed point problem for a suitable operator and derive some preliminary properties. We define the following operator a'IjJ
= v. ax
T'IjJ
- a(x, v)'IjJ(x, v) ; 'IjJ E D(T)
with domain
= {'IjJ E LOO(n x V);
D(T)
v. ~~ E LOO(n x V), 'ljJlr +
= O} .
In view of (10.6), (0 - T)-1 exists and
11(0 - T)-11IL(L<Xl(fl XV))
~
;*.
We note, in view of (10.5), that the right-hand side of (10.1) is equal to
-a (v)
t iv(k
Ck(X,
v,
k=1
v~, .. ., v~) [1 - IT (1- 'IjJ(x, V~))l'
(10.7)
3=1
Thus, we can write (10.1) in the form -T'IjJ = a(v)
t k=1
(k
iv
Ck(X,
v,
v~, ... , v~) [1- IT (1 -'IjJ(x, V~))l dv~ ... dv~ 3=1
~
with the condition 0
'IjJ
~
1. Introduce the set
B = {'IjJ E LOO(n x V); 0
~
'IjJ
~
I}
and the nonlinear operator in LOO(n x V) N: 'IjJ
--t
a(v)
t iv(
k
k=1
Ck(X,
v,
v~, ... , v~) [1 - IT (1 - 'IjJ(x, V~))l dv~ ... dv~. 3=1
We may formulate (10.1) (10.2) in operator form
-lim inf a(v)} v-+o
consists, at most , of isolated eigenvalues with finite multiplicities. Moreover, if this point spectrum is not empty then there exists a leading eigenvalue. The latter result can be proved directly or derived by duality from the corresponding result in Ll (0, x V) which follows from general results on positive co-semigroups (Theorem 5.2) . We say that Problem (10.8) is sub critical (resp. critical, resp. super critical) if the spectral bound
s(T)
:=
sup {Re..\; ..\
E
a(T + KH
225
Chapter 10. Nonlinear problems
is negative (resp. equal to zero, resp. positive). It is easy to see that this amounts to T(1 [(O-T)-lK] < 1 (resp. T(1 [(O-T)-lK] = 1, resp. T(1 [(0 - T)-1 K] > 1). We first give nonexistence results. We start with the easiest one.
10.4
The subcritical case
Theorem 10.2 If T(1 [(0 - T)-l K] < 1 then (10.8) has no nontrivial solution.
PToof Let
ep=Nep; epEB , ep=rfO, i.e. Using the decomposition N = K - L and the fact that L is non-negative,
(O-T)-lKep?,ep; ep?,O, ep=rfO which implies that
10.5
T(1
[(0 - T)-l K] ?, 1.
0
The critical case
We give here another nonexistence result which is more involved than the one above. We introduce the assumption I
:3 2::S: ko ::s: m;
I
Ck o (X , V,v 1 , ... ,Vko)
> 0 on
n x V k +1 0
•
(10.20)
Then
Theorem 10.3 Let T [(0 - T)-l K] = 1. If (10.20) is satisfied then (10 .8) has no nontrivial solution. (1
Proof Assume there exists a nontrivial solution '1jJ '1jJ ?, 0 , '1jJ =rf 0 on
n x v.
We note that ko
ko
j=l
j=l
II (1 - '1jJ(x, v~)) -1 + .L ep(x , v~) ?, 0 on n x Vk
Topics in Neutron Transport Theory
226
and Fko(z), defined by (10.13), vanishes only ifzi = 0 for alli E [1,2, ... , koJ. It follows, from (10.20), that
is strictly positive and consequently L'lj; > 0 a.e. on N'lj; < K'lj; a.e. on
n x V. Hence
n xV.
(10.21)
Let To and Ko be the operators in Ll(nxV) such that TO' = T and KG = K . Since Ko(O-To)-1 is power compact in Ll(n x V), and irreducible because [Ko(O - TO)-1]2 is positivity improving in view of (10.20), then there exists a strictly positive 'lj;o E Ll(n x V) such that
because ru
[Ko(O - TO)-I] = ra [(0 - T)-1 K] .
Hence, using (10.21) ,
where (., .) is the pairing between Ll(n x V) and Loo(n x V). This ends the proof. 1 then (10.8) has at least one non-
trivial solution. Proof. We recall that k
Fk : (Zb ..., Zk)
E
[0, IJk
-t
1-
IT (1 -
Zj)
j=1 and note that
Fk(O) = 0 ,
~~: (0) =
1 (1 SiS k).
(10.22)
227
Chapter 10. Nonlinear problems Thus there exists ~ E [0, Ilk, 0 ~ ~i ~ Zi such that k
8F k
H(z) = LZi 8z. (~). i=l
(10.23)
•
On the other hand, for each 0 < 1-£ < 1, there exists co
> 0 such that
so that, in view of (10.22) ,
8Fk -(z) 8zi
~
8Fk (1- 1-£)-(0) . 8zi
and, in view of (10.22) (10.23),
. k 8Fk k Fk(Z) ~ (1- 1-£) LZi 8z (0) = (1- 1-£) LZi for 0 ~ Zi ~ co (1 ~ i ~ k), i=l'
i=l
i.e.
1-
k
k
j=l
i=l
II(I- Zj) ~ (1- 1-£) LZi
for 0 ~ Zi ~ co (1 ~ i ~ k) .
(10.24)
Let 'ljJ* be a non-negative eigenfunction of (0 - T)-l K corresponding to its spectral radius and such that 11'ljJ*lluX>C!1xv)
= 1.
We define cp = c'ljJ* with c ~ co.
lt follows, from (10.24), that
1and then
k
k
j=l
j=l
II (1- cp(x, v~)) ~ (1 - 1-£) L cp(x, v~) , (1 ~ i ~ k)
(10.25)
Topics in Neutron Transport Theory
228 Hence
Nrp? (1- J-L)Krp and
Nrp? (1- J-L)(O - T)-1 Krp = (1 - J-L)ra [(0 - T)-1 K] rp.
(10.26)
We choose J-L in such a way that (10.27) Therefore Nrp ? rp, i.e. rp is a subsolution. We define inductively a sequence {rp d by rpo = rp and rp k+1 = Nrp k . By arguing as in the proof of Theorem 10.1, one verifies that {rpd C B is nondecreasing and converges in LOO(0. x V) to '1/;, a nontrivial fixed point of N. To prove that this solution is the minimal solution we need a technical result. Lemma 10.5 The solution given by Theorem 10.4 is independent of c. Proof Let 0 < Cl < C2 < co and let '1/;1, '1/;2 be the corresponding solutions. We set rpl = Cl '1/;* , rp2 = c2'1/;*·
From the order relation rpl :S rp2 and the construction of the solutions, it follows that '1/;1 :S '1/;2. Assume momentarily that (10.28) then '1/;1 is an upper bound of the inductive sequence which gives the solution '1/;2 and consequently '1/;2 :S '1/;1 and this implies the equality of the solutions. Thus it suffices to prove (10.28) . Define the set
E = {c; 0 < c :S C2, c'l/;* :S 'l/;1} . We note that E is a closed and nonempty interval (cl E E). Let c* be the least upper bound of E. Assume that c* < C2. By using (10.26),
(1 - J-L)ra [(0 - T)-1 K] x (c*'I/;*)
= (1 - J-L)(O - T)-1 K(c*'I/;*)
:S N(c*'I/;*) :S N('I/;I) = '1/;1' Therefore
c*(l-J-L)ra [{O-T)-IK] '1/;* :S'I/;1 which contradicts the definition of c· in view of (10.27). Thus c* hence (10.28) follows. We are ready to prove
= C2
and
229
Chapter 10. Nonlinear problems
Theorem 10.5 We assume that ru [(0 - T)-1 K] > 1 and that
:1(~):=.inf(X'VI)EnX vCl(X'V'~/»O {
onV
(10.29)
c(v):= inf(x,v)En x vCl(X,V,V) > 0 on V.
Then (10.8) has a nontrivial minimal solution. Proof Let us show that the nontrivial solution 'IjJ given by Theorem 10.4 is the minimal solution. Let cp be another nontrivial solution. To show that 'IjJ :S cp it suffices, according to lemma 10.5, to prove that
:3 c > 0 such that c'IjJ* :S cp
(10.30)
since cp would be an upper bound of the inductive sequence starting at c'IjJ* and tending to 'IjJ. We set Q
:= ru [(0 - T)-1
K].
Since and (10.31) it suffices to prove that
which is nothing but
In view of the positiveness of (0 - T)-I, it suffices that
Finally, this holds for c small enough if there exists c > 0 such that
Ncp 2: c.
(10.32)
We note that NCP=a(v)tlk k=1 v
Ck(X,V,V~ , ... ,v~) [1- J=1 rr(1-cp(X'V~))l dv~ . .. dv~.
Topics in Neutron Transport Theory
230 Hence Nt.p ~ a(v)
, " 1 dv l = C 1 t.p Jv( C1(X,V,V1)t.p(X,V1)
and, in view of (10.31) ,
so that (10.33) On the other hand, C 11 (0 - T) -1 C 11 t.p =
1
"
"
H(x, x , v, v )t.p(x , v )dx dv J
I
f!xV
where H(x, x', v, v') is equal to
1
00
o
and
x-x' x - x' x - x' ' x - x' ,ds e- a (-,-)sa(v)c1(X, v, --)a(--)c1(X , - - , v )~
s
C1 (., ., . ) I
s
s
is extended by zero outside V. It follows that
roo e-a(-s-)Sh(-s-) x - x' ds sn = G(x -
2
I
x-x'
H(x, x, v, v ) ~),* Jo
where h(v)
s
= £:1 (v)~(v). cf(o -
, x)
>0
Thus
T)-lCft.p
~
in
G(x - x')<j)(x')dx' := ~(x)
Iv
where <j)(x') = t.p(x', v')dv'. We note that <j) f=. 0 since t.p f=. 0, and that ~(x) > O. Finally, a simple convolution argument shows that ~(.) is continuous on R n whence ~( . ) is bounded away from zero on 0 and therefore (10.32) is satisfied, in view of (10.33) .
10.7
On the uniqueness
This section is devoted to the uniqueness of nontrivial solutions. Uniqueness turns out to be tied to a geometric property of certain operators related to N. We start with a useful definition. Definition 10.1 A non-negative nonlinear operator A defined on B c LOO(O x V) is said to be I-concave if, for any '1jJ E B, '1jJ f=. 0 and 0 < t < 1, there exist a, {3, J..L > 0 such that A(t'1jJ) ~ (1 {
a
+ J..L)tA('1jJ)
'5: A('1jJ) '5: {3.
231
Chapter 10. Nonlinear problems
We introduce the nonlinear operators Ck (1 ~ k ~ m) on LOO(0. x V) Ck 'ljJ
r
=
IT(V)Ck(X, v ,
iV k
v~, ... , v~) [1 - IT (1 - 'ljJ(x , V~))l dv~ . . . dv~ j=l
and give a preliminary uniqueness result. Theorem 10.6 If the operators CkN (1 ~ k ~ m) are I-concave, then (10 .8) has at most one nontrivial solution.
(3L
J.L1) Proof Let'ljJ1 and 'ljJ2 be two nontrivial solutions of (10.8). Let (01, the I-concavity of CkN and
= 1, 2) be the parameters corresponding to 'ljJi (i = 1,2) . We have (i
-
1
Ok
2
Ok
-
Ok
Ck'ljJ1 = Ck N 'ljJ1 2: Ok = {3~{3k 2: {3~CkN'ljJ2 = {3~Ck'ljJ2
which shows that the set
is not empty. Let tk := sup {t
> 0; C k'ljJ1 2: tCk'ljJ2} > O.
We first prove that (10.34) We argue by contradiction. Assume there exists some integer k such that tk = inf {t j ; 1 ~ j ~ m}
< 1.
Clearly Cj 'ljJ1 2: tkCj'ljJ2 ; 1 ~ j ~ m .
We note that N = 2:::7'=1 Cj and 'ljJ1
=
(0 - T)-l
",m
L..JJ =l
C j 'ljJ1 2: tk(O _ T)-l ",m Cj 'ljJ2 L..JJ =l
= tkN'ljJ2 = tk'ljJ2 · Thus N'ljJ1 2: N(tk'ljJ2) and Ck'ljJ1
= Ck N 'ljJ1
2: Ck N (tk'ljJ2)
2: (1 + J.L%)t kCkN('ljJ2)
232
Topics in Neutron Transport Theory
This contradicts the definition of tk. Thus, (10.34) holds and consequently
Hence
'l/J1 = (0 -
T)-l
2::=1 Ck'I/J1
~ (0 -
T)-l
2::=1 Ck'I/J2
=
'l/J2.
Changing the role of 'l/J1 and 'l/J2 yields the inequality 'l/J2 ~ 'l/J1 ' 0 To prove that CkN (1 ::; k ::; m) are I-concave, we need a technical result. Lemma 10.6 Let'I/J E B ,'I/J =I- O. Then, for all 0 < t < 1,
and is not identically zero. Moreover, if 'I/J
n x Vk,
> 0
a. e. on
nxV
then, on
Proof. Let Fk(Z) be defined by (10.10). We note that it vanishes at Z
= 0 and that
Let z E [0, I]k be fixed. We set h(t) := Fk(tZ) ; t E
[0, 1]
and note that
is nonincreasing in general and is decreasing if z > 0 (Zi > 0, Vi) . Using h(O) = 0 th' (t) - h(t) t2
233
Chapter 10. Nonlinear problems
th' (t) - (h(t) - h(O)) t2
th' (t) Hence (¥)' ~ 0 and, if general,
Z
>0
(Zi
~/h' (Bt)
(0
< B < 1).
> 0, Vi), (¥)' < 0 on )0, 1) . Thus, in (10.35)
and, if
Z
> 0, Fk;tZ) > Fk(Z) ;
t E
)0, 1[ .
This ends the proof by choosing Zi = 'lj;(x, v~) (1 ~ i ~ k). We complement the previous uniqueness result by
(10.36)
0
Theorem 10.7 We assume that (10.29) is satisfied and there exist Ak > 0 (2 ~ k ~ m) such that
then CkN (1 ~ k ~ m) are I-concave . Proof We begin with CIN. Let 'lj; E B, 'lj; =1= 0 and 0 < t < 1. We have to prove the existence of Ql, f31, III > 0 such that
(10.37) The existence of f31 is clear. We have m
N('lj;) = (0 - T)-1
L Ck'lj; k=1
so that
C 1 N('lj;) ~ C 1 (0 - T)- I C1 'lj;.
The arguments used in the proof of Theorem 10.5 show that C 1 (0-T)-lC1'lj; has a positive lower bound Ql . The first part in (10.37) amounts to
Topics in Neutron Transport Theory
234
Hence it suffices to prove the existence of d l
> 0 such that
cdJ"(t'l/J) - tClN('l/J) ~ dl and to choose III small enough. By linearity of Cl
(10.38) ,
m
ClN(t'l/J) - tClN('l/J) =
L
[Cl(O - T)-lCk(t'l/J) - tCl(O - T)-lCk('l/J)].
k=2 Moreover, Cl(O - T)-lCk(t'l/J) - tCl(O - T)-lCk('l/J) is equal to Cl(O - T)-l [Ck(t'l/J) - tCk('l/J)]
and Ck(t'l/J) - tCk('l/J) is equal to
/Vka(v)ck(X,V,V~, .. ., v~) [1-Il~=1(1-t'l/J(x,v~))] { - / Vk a(v)ck(X, v, v~, .. ., v~)t [1 Therefore
(10.39)
Il~=l (1 -
'l/J(x, v~)) 1 ~
o.
ClN(t'l/J) - tClN('l/J) ~ 0
in view of Lemma 10.6 and, for all k = 2, ... , m,
ClN(t'l/J) - tClN('l/J) ~ Cl(O - T)-l [Ck(t'l/J) - tCk('l/J)] .
(10.40)
Set
cp(x,v~, .. .,v~)= [1- rr(1-t'l/J(X'V~))l-t[1- rr(1-'l/J(X'v~))l · 3=1
3=1
We note, in view of Lemma 10.6, that cP side of (10.40) is equal to
=
/
n xvk
~
0, cp
i= O.
Thus, the right-hand
" , " , " , P (x" x ,v,vl' ... 'vk)cp(x ,vl , ... ,vk)dx dvl ·· · dvk
where P(x, ,x' , v, v~, ... , v~) is equal to 00
I
X- X ) e- u ( - . - Sa(v)cl(X, ~/
o
I
I
x-x x-x v, - - ) a ( - - ) s
s
(10.41)
235
Chapter 10. Nonlinear problems and
C1,
ck are extended by zero outside V. We note that
, , , P(x"x ,v,v1 "",vk) G(x-x'»O. Hence
where
\[!(x')
=
r cp(x',v~, ... ,v~)dv~ .. . dv~.
iVk Since \[! =1= 0, it follows that In G(x -
x')\[!(x')dx' > 0 (and is continuous by a convolution argument). Finally,jIO.38) follows in view of (10.40). We consider now the I-concavity of CkN for k ;::: 2. Let 'l/J E B, 'l/J =1= 0 and o < t < 1. We look for O!k , Ok , /-Lk > 0 such that CkN(t'l/J)
> (1 + /-Lk)CkN('l/J) (10.42)
The existence of Ok is clear. On the other hand, k
CkN('l/J) = Ck(O - T)-l
L Cj'l/J ;::: Ck(O - T)-lC 'l/J 1
j=l and Ck(O - T)-lC 1 'l/J is equal to
r a(v)ck(X, v, v~, ... , v~) [1- j=l IT (1- (0 - T)-lC 'l/J(x, V~))l, dv~ '" dv~ 1
iVk
so that
C,N(.p)
?y J.,
L
[1-
g
1
(1 - (0 - T)-'C,1>(x, v;)) dv; .. dv;
By the normalization assumption
IVk dv~ . .. dv~ =
1,
236
Topics in Neutron llansport Theory
where
p(x) =
L(o
- T ) - ~ C I $ ( vX1,) d v ' .
Arguing as previously, one sees that p ( x ) has a positive lower bound, whence the existence of a k follows. To prove the first part of (10.42), it suffices to prove the existence of dk > 0 such that:
We note, in view of (10.39),that
We set
k
p := (0 - T ) - l
>
z ~ j (( 0 $ -T ) )-'c~$.
j=1 Then
>
c k N ( t $ ) - t ~ k f i ( $ ) C k ( t p ) - tCk ('PI.
On the other hand, C k ( t p )- t C k ( p ) is equal to
and therefore
where @ ( x ,v ; , ...,v ; ) is equal to k
1-1
-(
j=1
Thus, using the fact that
X V ) )
(10.44)
237
Chapter 10. Nonlinear problems is nondecreasing and (10.44), we obtain
with
cp(x) =
Iv
(0 - T)-lC1'l/;dv .
As in the previous case, one sees that cp is bounded away from zero and therefore (10.43) is satisfied. 0 be fixed. Using the contraction property of { S ( t ) ;t 2 O),
and
where M is the Lipschitz constant of the polynomial operator N o n B since cp(s), $ ( s ) E B. Thus, we get existence and uniqueness on the time interval [o, Since the lifetime does not depend on the initial data in B, we can extend the solution by standard arguments. This ends the proof. 0
$1.
10.9
Time asymptotic behaviour
This section deals with the study of the limit as t -+ +oo of the time dependent solutions. To this end we give another approach of (10.46) based on monotonicity arguments. Theorem 10.9 Let cp E B and define the inductive sequence {+k)
(i) If cp is a subsolution, i.e. cp 5 %cp, then {$k) is nondecreasing and converges pointwise to the solution of (10.46) . (ii) If cp is a supersolution, i.e. cp 2 then {$k) is nonincmasing and converges pointwise to the solution of (10.46).
rep,
239
Chapter 10. Nonlinear problems
Proof We note that
We already know that N leaves B invariant. By using the fact that N is nondecreasing and that S(t) is non-negative the claim follows by standard arguments used previously. Before proceeding further we give a preliminary result. Lemma 10.7 The subspace Xex> of X consisting of those maps having a limit in Lex>(n x V) norm as t -+ +00 is invariant under N. More precisely, if cp E Xex> then Ncp E Xex> and lim Ncp(t) = (0 - T)-l [ lim cp(t)]. t-++ex> t-++ex>
Proof Let cp E Xex> and let CPex> = limt-++ex> cp(t). We have Ncp(t) = S(t)'l/Jo We note that S(t)'l/Jo
J:
-+
+ lot S(t - s)Ncp(s)ds .
0 in Lex> as t
S(t - s)Ncp(s)ds
=
-+
J:
+00 and
S(s)Ncp(t - s)ds
= J~ X[O,t] (s)S(s)Ncp(t - s)ds . In view of the continuity of N in Lex>(n x V), X[O,t] (s)S(s)Ncp(t
- s)
-+
S(s)N(cpex» ,
Moreover
loex>
IIS(s)11 ds < 00.
Hence, by the dominated convergence theorem,
and this ends the proof.
\:j
s > o.
240
Topics in Neutron Transport Theory
Theorem 10.10 If ru [(0 - T)-l K] < 1, or if ru [(0 - T)-l K] (10 .20) holds, then 'ljI(t) --+ 0 in LOO(O x V) as t --+
=
1 and
+00
where'ljl(.) is the solution of the Cauchy problem (10.46) . Proof. We define the inductive sequence {¢d, ¢k+l = N'ljIk, ¢o = 1. Then, according to Theorem 10.9, {¢d is nonincreasing and converges pointwise to 'ljI( .). Moreover, according to Lemma 10.7,
It follows, by induction, that
(10.47)
where {1pd is defined inductively by
One sees that {~d is nothing but the sequence used in Theorem 10.1 to prove the existence of a maximal solution to the stationary problem, and
where
~
is the maximal solution. Hence
and, in view of (10.47) ,
This ends the proof because ~ = O. The treatment of the supercritical case is more technical, and we need a preliminary result. Let 'ljI* be a non-negative eigenfunction of (0 - T)-l K corresponding to its spectral radius introduced in Section 10.6, normalized by 11'ljI* IILOO(O XV ) = 1 and 'Po = c'ljl*, with c::; co (see (10.25)) . According to (10.26), (10.48) We have
241
Chapter 10. Nonlinear problems Lemma 10.8 Let ra [(O-T)-lK]
> 1. We assume that the initial data
1/Jo of the Cauchy problem (10.46) is bounded away from zero. Let z = inf 1/Jo· 1 If c ~ ).* IIKII- z then CPo ~ Ncpo, i.e. CPo is a subsolution of N. Proof Let us prove that N CPo - CPo 2: 0, i.e.
+ lot S(s)Ncpods - CPo 2: O.
S(t)1/Jo
According to (10.48), it suffices to show S(t)z + lot S(s)Ncpods - (0 - T)-l Ncpo 2: 0 ,
i.e. S(t)z
-1
00
(10.49)
S(s)Ncpods 2: O.
Let (x, v) E n x V. For t > s(x, v), each term in (10.49) vanishes. If t < s(x, v), then (10.49) reduces to e-ta(v) z -
J~ e-sa(v) Ncpo(x + sv, v)xn(x + sv)ds
= J~ e-sa(v) On the other hand, N CPo
[u(v)z - Ncpo(x ~
+ sv, v}xn(x + sv)] ds.
K CPo and
u(v)z - cK1/J*(x + sv, v) 2: )'*z - c IIKII 2: 0
and the proof is complete.
Theorem 10.11 Let ra [(0 - T)-l K] > 1. We assume that the initial data 1/Jo of the Cauchy problem (10.46) is bounded away from zero and that the nontrivial solution of the stationary problem V5 is unique, then
where 1/J(.) is the time dependent solution. Proof We define the inductive sequence {1/J k }, 1/J k -
-
+1
= N1/J-k'1:..o .1. = CPo .
{tk}
According to Lemma 10.8, CPo ~ Ncpo. By using Theorem 10.9, is non decreasing and converges pointwise to 1/J(.). According to Lemma 10.7,
t1(t)
--t
(O-T)-lNcpo in LOO(n x V).
Topics in Neutron Transport Theory
242 It follows, by induction, that
where the sequence {!ek}' defined inductively bY!ek+l
= (O-T)-lN!ek ':£0 =
'Po, is nothing but that used in Theorem 10.4 to prove the existence of a
nontrivial solution. Thus
where (~d is the sequence introduced in the proof of Theorem 10.10. Hence
111fJ(t) - 0
(11.7)
S E V(]a, b[ x ]-1 , +1[; dx ® d la!) .
The following evenness assumptions are crucial for the sequel
(11.8) and
S(x, Ik) = S(x, -Ik) dial a.e.
(11.9)
We state the elementary result Proposition 11.1 We assume that condition (11.7) is satisfied. (11.4) has a unique solution in W given by (11 .5) .
Then
The main result is the following Theorem 11.1 Let (11.6)-(11.9) be satisfied and let 7/J denote the solution of (11 .4). We assume that
is finite. Then
(i) 7/J E V(]a, b[ x ]-1, +1[ ;dx ® ~)
249
Chapter 11. Velocity averages and inverse problems
(ii) IPl(') =
+1 /
-1
'f/;(.,/J)da(/J) E W 2,P(]a , bD and
Before giving the proof, let us derive some direct consequences.
Corollary 11.1 Let the conditions of Theorem 11.1 be satisfied. If S(x, /J) = +1
Sl(X)S2(/J) and if.[
-1
a(/J)S2(/J)d:\f)
I- 0,
then
where
Remark 11.1 Let d{J =
a( 1L)2
~da .
knowledge of the moments
/
explicitly Sl(X) provided that
Then Corollary 11.1 expresses that the
+1 'f/;(x, /J)da(/J) -1
and
/+1 'f/;(x, /J)d{J(/J) yields -1
+1 /
a(/J)S2(/J)do:\j') is known. Note that the -1
!Jo
determination of Sl does not depend explicitly on the incoming distribution (g-,g+) . The choice of evenly spaced Dirac measures yields
Corollary 11.2 Let /Jo E ]0, 1[ and let da = D!Joo
+ D-!Joo'
Then
2
S(x, /JO) = [IP2(X) where IPl(X) = 'f/;(x,/Jo)
+ 'f/;(x, -/Jo)
IP~ (x)] 2~~o)
and IP2(X) = ~IPl(X). !Joo
(11.10)
Topics in Neutron Transport Theory
250
Remark 11.2 As a consequence of Corollary 11.2, one sees that, if 0'(.) and S( x , .) are even then tp(x, v) = 1jJ(x , v) + 1jJ(x, -v) satisfies the second order equation
Thus we find again the even formulation of the transport equation [30] . Proof of Theorem 11.1 : The part (i) is a simple consequence of Proposition 11.1 where we use the measure ~ instead of dial. Consider now part (ii) . The solution 1jJ is given by (11.5) so that, by the evenness assumption, tp1(X) is equal to
J J 1
0
1jJ(x, Ji-)da(Ji-)
+
o
J
1jJ(x, Ji-)da(Ji-)
-1
1
o
+
e-lW(b-x)g_(Ji-)da(Ji-)
-1
J Je-l'Wlx-x'IS(x"Ji-)da~Ji-) b
+
J 0
e-lW (x-a)g+ (Ji-)da(Ji-) 1
dx'
a
.
(11.11)
0
The last term is given by
It follows that tp~ (x) is equal to
-I: (1~e)e-l'W(x-a)g+(Ji-)da(Ji-) +
J:
-I:
S(X,Ji-)daje) -
S(x , Ji-)daje)
+
+ J~l
'~l)e-lW(b-x)g_(Ji-)da(Ji-)
J: J: '~l)e-lW(x-x')S(X',Ji-)d,jr) J: J: (11~l)e-lW(x'-x)S(X" Ji-)d,jr) · dx'
dx'
251
Chapter 11. Velocity averages and inverse problems Differentiating again, one sees that O, the solution of (11 .14) is given by r(x,v)
'IjJ(x, v)
= Jo
e-sa(v)S(x-sv)ds
where
s (x, v) = inf {s >
(11.16)
°;
x - sv tJ. n} .
Let df.L(v) = da(p) ®dT(W) be a (signed) measure on V where dT(W) is the surface Lebesgue measure on the unit sphere sn-l and da(p) is a (signed) measure on [0,1) satisfying the conditions
1 1
da(p) =
o
° 11 dial ,
PoP
(p)
2
O± and to the size of their spectral radii. Moreover, it will turn out that the above problems are equivalent. We point out, in view of the boundedness of {Uo(t); t E R}, that
aCTo) C iR. On the other hand, a(To)nR (see [27] p. 295) . Hence
=1=
0 in view of the positiveness of {Uo(t); t
E
R}
o E aCTo) . Thus s limA--->o± ().. - TO)-1 do not exist in L1(J1-). One may interpret the existence of s limA--->o± B().. - TO)-1 as a limiting absorption principle where slimA--->o±().. - TO)-1 would exist as operators from L 1(J1-) into a larger space. Positivity as well as the choice of P space are crucial assumptions because of the additivity of the L1 norm on the positive cone. We already saw the importance of such assumptions in Chapters 8 and 9. We will
270
Topics in Neutron Ti-ansport Theory
apply the abstract results to transport operators in the whole space where {Uo(t);t E R) represents the streaming group and B the collision operator, and show how the relevant abstract assumptions are expressed naturally in terms of properties of the different cross-sections.
12.2
Preliminary results
We begin with the following observation. Let X = L1 ( p ) .
Lemma 12.1 Let { W ( t ) ;t > 0 ) be a bounded positive q-semigroup on X with generator G and let C E L+(L1(p);L 1 ( p ) ) . Then (2)
lo rt
lim C t-+m
W ( s ) x d s exists for all x E X
(12.1)
i f and only i f Limx,o+ C(X - G ) - l x exists for all x E X . I n that case, we have
W ( s ) x d s = C(O+ - G ) - l x ; x E X
(12.2)
where C(O+ - G)-l denotes the strong limit s lirn~,~, C(X - G)-l (zi) Similarly, i f { W ( t )t; E R ) is a bounded positive q-group, then lim
t-+-m
c J!
W ( s ) x d s exists for all x E X
(12.3)
i f and only i f limx,o- C ( A - G ) - l x exists for all x E X . I n that case, we have
W ( s ) x d s = -C(O- - G)-'x ; x E X .
(12.4)
Proof: Since the positive cone L i ( p ) is generating (i.e. L1 ( p ) = L i ( p )L i ( p ) ) ,we may restrict ourselves to x E L i ( p ) . Let the strong limit (12.1) exist and denote it by C J : ~W ( S ) Z ~Then, S . for X > 0 ,
Since C(X - G)-l is nondecreasing in X > 0 it follows, by the monotone convergence theorem, that limx,o- C(X - G ) - l x exists and
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Chapter 12. Limiting absorption principles
Conversely, assume that C(O+ - G)-l exists. From the obvious inequality
we get
Letting>.
-t
0+, we obtain
By the monotone convergence theorem, we obtain the converse result and the reverse inequality
(+oo
C io
W(s)xds ~ C(O+ - G)-Ix .
This ends the first part of the Lemma. To deal with the second part, we where W(t) = introduce the positive bounded co-semigroup {W(t); t ~
O}
-
+00-
W( -t), with generator G = -G. According to the first part, C Io W(s)ds exists if and only if C(O+ - 0)-1 exists and these strong limits are equal. This amounts to
C
1°00 W(s)xds = -C(O_ - G)-Ix;
xEX
and the proof is complete. 0 The boundedness of the perturbed (semi)group plays a key role for the existence of wave operators. This point is clarified in the following Theorem 12.1 (i) We assume that the strong limit
B(O+ - TO)-l
:= s lim >'-.0+
B(>. - TO)-l
exists and that ru [B(O+ - TO)-l] < 1. Then T = To bounded positive co-semigroup {U(t); t ~ O} . (ii) If, in addition,
B(O_ - TO)-l exists and ifru [B(O_ - TO)-l] co-group {U(t); t E R}.
:= s lim >.-.0_
+B
generates a
B(>' - TO)-l
< 1, then T = To + B generates a bounded
272
Topics in Neutron a m s p o r t Theory
To prove this theorem we need a technical preliminary result. We define the space H L of strongly continuous mappings
z : t E [O, +oo[ + Z ( t ) E L ( L ~ ( P L) ;~ ( P ) ) such that suptlo IIZ(t)ll < +oo, endowed with the norm
llzll := St Ul 0P IIZ(t)ll .
(12.6)
It is easy to see that H& is a Banach space for the norm (12.6). We introduce the operator
where the integral (12.7) is understood as a strong one. We prove first
Lemma 12.2 The operator L is bounded in H& and
ru(L) 5
[B(o+- ~ o ) - l ].
Proof: Let x E X+ and Z E H&. From
we derive the estimates
In view of the additivity of the L1 norm on X+,
I l B ~ o ( ~ ds ) ~= ll
1 Jot
~uo(s)xdSII.
Hence, using Lemma 12.1,
= llZll IIB(o+ - TO)-~XII I llZll IIB(O+ - To>-lII llxll.
If x E X is arbitrary, by decomposing it into positive and negative parts, IILz(t>xllI llzll IIB(O+ - ~ 0 ) - ~ ~ ~ [ 1 + 1 ~11+ 2-11 111 = IlZll IIB(o+ - To)-lIJ llxll .
Chapter 12. Limiting absorption principles This shows that LZ E H& and IILIILc
HL) I IIB(O+ - T ~ ) - l ( l .
For each integer n, a computation shows that LnZ(t) is equal to
Hence, for x E X+, I(LnZ(t)xll is less than or equal to
On the other hand, the latter is less than or equal to
which is equal to
where the additivity of the norm on X+ is used. The change of variables
shows that
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274
so that, for arbitrary x E X,
IILn Z(t)xll is less than or equal to
Hence and consequently
Finally
r t1 (L) ::; r t1 [B(O+ - TO)-l] and this proves the claim. ProoJofTheorem 12.1: Since Uo(.) E
U(t)
=
Uo(t)
+
H~,
the Duhamel equation
lot U(s)BUo(t - s)ds ,
written abstractly as
U-LU= Uo , has a unique solution U E
H~
given by
because r t1 (L) ::; r t1 [B(O+ - TO)-l] < 1. Thus {U(t); t 2: O} is bounded. This ends the first part of the theorem. The second part is dealt with similarly by introducing
Uo(t) = Uo( -t) , U(t) = U( -t) (t 2: 0) and noting that
-
-
U+LU= Uo where
L: Z E H~ ~ lot Z(s)BUo(t -
One proves, as in Lemma 12.2, that
and we proceed as in the first part.
s)ds.
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Chapter 12. Limiting absorption principles
12.3
On the wave operators
This section is devoted to the existence of the following strong limits
limt. . . . s limt . . . . s
+oo
U(t)Uo( -t)
+oo
Uo(-t)U(t)
s limt ....... -
oo
U(t)Uo(-t).
We begin with Theorem 12.2 If both B(O± - TO)-l exist and ifru [B(O+ - TO)-l] then s limt ....... +oo U(t)Uo( -t) exists.
< 1,
~
O} is
Proof: According to the first part of Theorem 12.1, {U(t); t bounded. We set
IIUII =
sup IIU(t)ll · t~O
From
U(t) = Uo(t)
+
lot U(s)BUo(t - s)ds
one sees that
U(t)Uo(-t)x
=
x+
lot U(s)BUo(-s)xds.
Let x E X+ . Then
I:
IIU(s)BUo( -s)xll ds
~ IIUII l~ IIBUo(-s)xll ds = 'IU"III~ BUo( -S)xdsll = "u"III~oo BUo(S)Xdsll = IIUIIII-B(O- -
TO)-lxll ·
Hence limt ....... +oo U(t)Uo( -t)x exists for all x E X+ and finally for all x E X since X = X+ - X+. 0 Symmetrically we have the following Theorem 12.3 If both B(O± - TO)-l exist and ifru [B(O_ - TO)-l]
then s
limt. . . . -oo U(t)Uo( -t) exists.
< 1,
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276
Pr0o.f According to the second part of Theorem 12.1, { f i ( t )t,
> 0 ) is
bounded ( 6 ( t )= U ( - t ) ) and
-
where Uo(t)= Uo(-t), i.e.
Hence
6(s)BUo(s)xds; t 2 0.
f i ( t ) ~ o ( t )= xx By choosing z E X+,
Thus s limt,-, U(t)Uo(-t) exists because of X = X+ - X+. 0 The analysis of s limt-.+ooUo(-t)U(t) requires a technical preliminary result.
Lemma 12.3 The following assertions are equivalent. (i) B(O+ - To)-' exists and r, [B(O+- To)-'] < 1. (iz) (0,oo) c p(T), (A-T)-' is positive ( A > 0 ) and B(O+ -T)-'exists. We omit the proof which is quite similar to that of Lemma 8.1. We observe however that we have no similar result for B(0- - T)-' because U ( t ) need not be positive for t < 0 and (A - T)-' fails to be positive for X negative (and large). We are now ready to show
Theorem 12.4 If B(O+ - T)-'exists and if r, [B(o+- To)-']
< 1, then
s lim Uo(-t)U(t) exists. t-+m Proof: According to Lemma 12.3, B(O+ - T)-' exist or, equivalently (see Lemma 12.1), rt
s lim BU(s)ds exists. t-+m Jo
277
Chapter 12. Limiting absorption principles Thus
U(t) = Uo(t)
+
lot Uo(t - s)BU(s)ds
yields
Uo( -t)U(t)x = x +
lot Uo( -s)BU(s)xds.
By choosing x E X+ and noting that U(s) is positive for s :::: 0,
I:
IIUo( -s)BU(s)xll ds
~ sups~o IlUo(s)11 J: IIBU(s)xll ds =
sups~o IlUo(s)IIIIJ: BU(s)XdSIl
~ sups~o IIUo(s)IIIIJ~ BU(s)Xdsll = sups~o
This ends the proof since X
12.4
IlUo(s)IIIIB(O+ - T)-lXII ·
= x+ - x+.
The similarity of To and T
We show how the existence of wave operators provides a useful tool to show the similarity of the generators of the groups under consideration. Theorem 12.5 Let both B(O± - TO)-l exist and ru [B(O± - TO)-l] Then the following wave operators exist
W+(To, T)
:= s limt-++oo
Uo(-t)U(t)
W+(T, To) = s limt-++oo U( -t)Uo(t). Moreover,
and
0 such that IIUo(t)xll 2 c 11x11 for all x E X and all t E R. We assume, from now on, that {U(t); t E R) is a positive and bounded CO-group.
Theorem 12.7 The following assertions are equivalent: (i) s limt,+, Uo(-t)U (t) exists (ii) {Uo(-t) U(t) ;t > 0) is bounded (iii) {U(t); t 2 0) is bounded. Proof: We note that (i)+(ii) by the uniform boundedness theorem and (ii)==+(iii) because {U(t); t E R} is bounded. Let (zii) be true. Then, according to Corollary 12.2, B(O+ -To)-' exists and r, [B(o+ - TO)-'] < 1, and therefore the strong limit slimt,+, Uo(-t)U(t) exists in view of Theorem 12.4. 0 We end this section with
Theorem 12.8 The following assertions are equivalent: (i) s limt,+, U(t)Uo(-t) exists (ii) {U(t)Uo(-t) ; t > 0) is bounded (iii) {U(t); t 2 0) is bounded. Proof: The parts (i)& (ii) and (ii)* (zii) are clear. Let (iii) be true. Then, according to Corollary 12.2, B(O+ - To)-' exists and
On the other hand,
implies
and, since U(t) 2 Uo(t) (t rt
Let x E X+, then
> O),
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Chapter 12. Limiting absorption principles Hence, using the property
IlUo(t)xll
~
c IIxll for all x E X and all
t E R,
::::: J:UUo(S)BUo(-s)xlldS ::::: sUPr2:0
IIU(r)Uo( -r)llllxll·
By the monotone convergence theorem, S
lim t--->+oo
Jot
BUo( -s)ds exists,
i.e., according to Lemma 12.1,
B(O_ - T)-l exists. Finally s limt--->+oo U(t)Uo( -t) exists, in view of Theorem 12.2. 'te -
J:
a(x-sv,v)ds 1jJ{x
- tv, v)dt (A> 0)
283
Chapter 12. Limiting absorption principles and
By choosing 'Ij;
~
0, the monotone convergence theorem yields
On the other hand,
J
dxd/1-(v)
J
00
b(x, v , v')d/1-(v') / e - Jo' u(x-sv' .V' )dS'Ij;(x - tv', v')dt
v
0
is equal to
J JJ 00
d/1-(v')
v
dt
0
h(x+tv' ,v')e- Jo'u(x+sv' .v')ds'Ij;(x,v')dx
(12.16)
Rn
which is less than or equal to
This shows that B(O+ - TO)-l exists and (12.15) holds under assumption (12.13) . Conversely, let B(O+-To)-l exist and let (12.12) hold, then (12 .16) shows that
(12.17) Thus, the left-hand side of (12.17) defines a continuous functional on
whence
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284
i.e. (12.13) holds. This ends the proof of the first part. To deal with the second part we observe that, under (12.12),
and we proceed as previously. The existence of wave operators for transport equations are direct consequences of the previous results. The results are summarized in the following Theorem 12.10 (i) Let (12.12) - (12.14) hold and let M+(h) < 1, then s limt-++oo U(t)Uo( -t) exists. (ii) Let (12.12) - (12.14) hold and let eM+(cr) M_(h) < 1, then the strong limit s limt--+-oo U(t)Uo( -t) exists. (iii) Let (12.13) hold and let M+(h) < 1, then s limt--++oo Uo( -t)U(t) exists.
This theorem takes advantage of the norms of the operators B(O± TO)-l while the relevant parameters are their spectral radii. We point out that B(O+ -TO)-l is positive and -B(O_ -TO)-l is also positive. To refine the above estimates we will assume, in the case where the spectral radius of B(O+ - TO)-l is positive (resp. the spectral radius of B(O_ - TO)-l is positive), that it is an isolated eigenvalue of B(O+ - TO)-l (resp. -B(O_TO)-l). This is true, for instance, if B(O± - TO)-l are power compact. Under this assumption we have Lemma 12.4 (i) Let (12.13) hold. We assume that
b+(v, v') = ess sup roo b(x + sv ' , v, v')ds xERn
Jo
is the kernel of a bounded operator B+ E L(Ll(V; dp,)). Then rcr [B(O+ - TO)-l] :S rcr(B+).
(ii) Let (12.12) and (12.14) hold. We assume that 00
I
L(v,v) = ess sup xERn
1
I
I
b(x - sv ,v,v )ds
0
is the kernel of a bounded operator B_ E L(Ll(V;dp,)). Then
285
Chapter 12. Limiting absorption principles Proof (i) Let ru [B(O+ - TO)-l] a corresponding eigenfunction
{
, , roo
Jvb(x,v,v)dJ.£(v)Jo
so that a1jJ(x,v):::;
i
e-
= a > 0 and
let 1jJ E L~(Rn x V) be
J.'oUx-sv,v ( ")d s1jJ(x-tv,v')dt=a1jJ(x,v) '
1+
00
b(x,v,v')dJ.£(v')
1jJ(x-tv',v')dt.
(12.18)
By integrating (12.18) with respect to x, we get acp(v):::;
i
dJ.£(v')
kn [1+
00
dx
b(x+tv',v,v')dt] 1jJ(x,v')
(12.19)
where cp(v) = (
JRn
1jJ(x,v)dx.
It follows from (12.19) that acp(v) :::;
i
b+(v, v')cp(v')dJ.£(v') = B+cp.
Hence ru(B+) ~ a = ru [B(O+ - TO)-l]
and this ends the proof of (i). To deal with (ii) we choose 1jJ E L~(Rn x V) such that where a = ru [-B(O_ - To)-~] > 0 and we proceed as in (i). 0 We complement Theorem 12.10 with Theorem 12.11 (i) Let (12.12) - (12.14) hold. If B(O+ - TO)-l is power compact and if ru(B+) < 1, then s limt-++oo U(t)Uo( -t) exists. (ii) Let (12.12) - (12.14) hold. If B(O_ - TO)-l is power compact and if eM+(u)ru(B_) < 1, then s lim t -+- oo U(t)Uo( -t) exists. (iii) Let (12.13) hold. If B(O+-To)-l is power compact and ifru(B+) < 1, then s limt-++oo Uo( -t)U(t) exists. Remark 12.2 Sufficient conditions ensuring the power compactness of B(O±TO)-l are given in [25]. In the context of Lebesgue measure (i.e. dJ.£(v) =
Ivl Iv'l
0 for ~ (and the cross-sections are compactly supported), then ru(B±) = 0 and therefore the wave operators exist regardless of the size (i.e. the norm) of B .
dv), if b(x,v,v')
=
286
12.7
Topics in Neutron nmsport Theory
Comments
The material in this chapter was taken from M. Mokhtar-Kharroubi 1251 where, more generally, unbounded (but To-bounded) collision operators B are considered. In the same spirit, an abstract two L1-spaces theory with applications to exterior problems (i.e. time asymptotic equivalence of the dynamics outside a bounded obstacle, with reflection at the boundary, and the free dynamics without obstacles) is given by M. Chabi, M. MokhtarKharroubi and P. Stefanov 191. We point out that the choice of L1spaces is a keypoint in this theory while positivity can be weakened (to some extent) by using domination arguments. However, for positive finite rank (or nuclear) perturbations, this theory was extended partially to general Banach lattices by M. Chabi and M. Mokhtar-Kharroubi [8]. The scattering theory for transport operators was initiated by J. Hejtmanek 1151, B. Simon [34] and developed by V. Protopopescu 1291, J. Voigt 1381, W. Schappacher [33] where Cook's method is used as well as positivity arguments. We mention a study of the range of wave operators by T. Umeda [36],by means of Enss decomposition principles. The wave operators for transport equations outside an obstacle (with reflection on the obstacle) as well as spectral theory are analyzed in P. Stefanov [35] (see also 191). Spectral results in the whole space are given in B. Montagnini 1261, J. Hejtmanek [16], A. Huber 1171, G. Greiner [14] and T. Umeda 1361. The Lax-Phillips formalism was applied to transport equations by H. Emamirad [lo] (see also H. Emarnirad [ll]).An abstract scattering theory in Banach lattices with applications to transport equations in L1 spaces was given by T. Umeda [37]. Nevertheless this theory does not apply, in general, to transport equations in P spaces (1 < p < co). We will show, in Chapter 13, that scattering theory for transport equations in P spaces (1 < p < co) is more suitably handled by Lin's factorization techniques [23]. Behind the existence of wave operators, in transport theory, there is the locally decaying property of transport operators which was pointed out by J. Voigt 1381. This property is tied to dispersive effects of velocity averages and will appear more transparently in Chapter 13. We point out a generalization of scattering theory to partly transparent surfaces (i.e. the dynamics in the whole space is subject to a transmission condition in some bounded region) by V. Protopopescu 1301. Useful relationships between the scattering operator and the albedo operator (for interior problems) are given in P. Arianfar and H. Emamirad [I],V. Protopopescu [31]and H .Emarnirad and V. Protopopescu 1121. The known literature on wave operators for transport equations is concerned only with bounded transport semigroups, referred to as subcritical or non-proliferating systems. This excludes, of course, the presence of point spectrum. At this point we mention an open (and probably difficult) problem : Assume that
Chapter 12. Limiting absorption principles
287
the cross-sections are compactly supported but the system is proliferating, giving rise to discrete point spectrum. Is it possible to define, in this case, generalized wave operators on a suitable subspace X I (with a reasonable description of X I ) ? The difficulty of the problem is tied, of course, to the lack of a reasonable functional calculus for transport operators. It is well known that the wave operators provide a mean to prove similarity results (see Theorem 12.5). This is known as the time dependent method of similarity. A stationary method (in Hilbert spaces) was given by T. Kato 1201 and extended (to Banach spaces) by S.C. Lin [24]. This method is applied to transport operators by M. Chabi [7]. At last, we point out that Runaway phenomena, in the kinetic theory of particle swarms, give rise to interesting scattering problems involving time dependent evolution problems (time dependent velocity) [2] - [6], [13]. It would be useful, and probably possible, to extend the abstract formalism of this chapter to time dependent evolution groups on L1(p) spaces. The aim of this chapter was merely to present one aspect of scattering theory (the existence of wave operators) tied to positivity in L1(p) spaces and motivated by transport theory. However, scattering theory is a very vast (and richer) subject for which we refer to the books [18] Chapter X, 1211, 1281 and the references therein.
References [I] P. Arianfar and H. Emarnirad. Relation between scattering and albedo operators in linear transport theory. Dansp. Theory Stat. Phys. 23(4) (1994) 517-531. [2] L. Arlotti. On the asymptotic behaviour of electrons in an ionized gas subject to time dependent electric field. Dansp. Theory Stat. Phys. 21(46) (1992) 733-752. [3] L. Arlotti and G. F'rosali. Long time behaviour of particle swarms in runaway regime. Op. theory: Adv. Appl. 51 131-143. Birkhauser Verlag, Basel, (1991). [4] L. Arlotti and G. F'rosali. Runaway particles for a Boltzmann-like transport equation. Math. Models Methods Appl. Sci. 2(2) (1992) 203-221. [5] G. Busoni and G. F'rosali. Asymptotic behaviour of a charged particle transport problem with time-varying acceleration field. Damp. Theory Stat. Phys. 21(46). (1992) 713-732.
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[6] G. Busoni and G. Frosali. Large time behaviour of drift velocity in a charged particle transport problem. Preprint. [7] M. Chabi. Similitude d'operateurs de transport. Methode stationnaire. Work in preparation. [8] M. Chabi and M. Mokhtar-Kharroubi. On perturbations of positive co-(semi)groups on Banach lattices and applications. J. Math. Anal. Appl. 202 (1996) 843-861. [9] M. Chabi, M. Mokhtar-Kharroubi and P. Stefanov. Scattering theory with two L1 spaces: application to transport equations with obstacles. Ann. Fac . Sci. Toulouse. (1997). To appear. [10] H. Emamirad. On the Lax and Phillips scattering theory for transport equation. J. Funct. Anal. 62 (1985) 276-303. [11] H. Emamirad. Scattering theory for linearized Boltzmann equation. Transp. Theory Stat. Phys. 16 (1987) 503-528. [12] H. Emamirad and V. Protopopescu. Relationship between the albedo and scattering operators for the Boltzmann equation with semitransparent boundary conditions. Math. Methods Appl. Sci. [13] G. Frosali, C.Van der Mee and S.L. Paveri Fontana. Conditions for runaway phenomena in the kinetic theory of particles swarms. J. Math. Phys. 30(5) (1989) 1177-1186. [14] G. Greiner. Spectral properties and asymptotic behavior of the linear transport equation. Math. Z. 185 (1984) 167-177. [15] J . Hejtmanek. Scattering theory of the linear Boltzmann operator. Comm. Math. Phys. 43 (1975) 109-120. [16] J. Hejtmanek. Dynamics and spectrum of the linear multiple scattering operator in the Banach lattice L1(R3 x R 3). Transp . Theory Stat. Phys. 8(1) (1979) 29-44. [17] A. Huber. Spectral properties of the linear multiple scattering operator in L1-Banach lattices. Int. Eq. Op. Theory 6 (1983) 357-371. [18] T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag, 1984. [19] T . Kato. Scattering theory. Studies in mathematics. Math. Assoc. Amer. 7 (1971) 90-115.
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[20] T. Kato. Wave operators and similarity for some non-self-adjoint operators. Math. Ann. 162 (1966) 258-279. [21] P.D. Lax and RS. Phillips. Scattering Theory. Academic Press, Inc, 1989. [22] P.D. Lax and RS. Phillips. Scattering theory for dissipative systems. J. Funct. Anal. 14 (1973) 172-235. [23] S.C. Lin. Wave operators and similarity for generators of semigroups in Banach spaces. Trans. Amer. Math. Soc. 139 (1969) 469-494. [24] S.C. Lin. A stationary approach to perturbation of operators in Banach spaces. J. Math. Anal. Appl. 32 (1970) 352-369. [25] M. Mokhtar-Kharroubi. Limiting absorption principles and wave operators on L1(J.-L) spaces. Applications to transport theory. J. Funct. Anal. 115 N01 (1993) 119-145. [26] B. Montagnini. The eigenvalue spectrum of the linear Boltzmann operator in Ll(Rn) and L2(Rn) . Meccanica. (1979) 134-144. [27] R Nagel (Ed). One-parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, 1184, Springer-Verlag, 1986. [28] V. Petkov. Scattering Theory for Hyperbolic Operators. North-Holland, 1989. [29] V. Protopopescu. On the scattering matrix for the linear Boltzmann equation. Rev. Roum. Phys. 21 (1976) 991-994. [30] V. Protopopescu. Une generalisation de la theorie de scattering pour les equations du trarISport lineaires. C. R . Acad. Sc. Serie 1. 317 (1993) 1191-1196. [31] V. Protopopescu. Relation entre les operateurs d'albedo et de scattering avec des conditions aux frontieres non transparentes. C. R. Acad. Sc. Serie 1. 318 (1994) 83-86. [32] M. Reed and B. Simon. Methods of Modern Mathematical Physics. Vol 3: Scattering Theory. Academic Press, 1979. [33] W. Schappacher. Scattering theory for the linear Boltzmann equation. Ber Math. Stat Sekt. Forschungszentrum. Graz. 69 (1976) . [34] B. Simon. Existence of the scattering matrix for the linearized Boltzmann equation. Comm. Math. Phys. 41 (1975) 99-108.
290
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[35J P. Stefanov. Spectral and scattering theory for the linear Boltzmann equation in Exterior domain. Math. Nachr. 137 (1988) 63-77. [36J T. Umeda. Scattering and spectral theory for the linear Boltzmann equation. J. Math. Kyoto Univ. 24(2) (1984) 205-218. [37J T. Umeda. Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators. J. Math . Soc. Japan 38(4) (1986) 617-625. [38J J. Voigt. On the existence of the scattering operator for the linearized Boltzmann equation. J. Math. Anal. Appl. 58 (1977) 541-558.
Chapter 13
Lin's factorization formalism and applications to transport theory 13.1
Introduction
This chapter is devoted to scattering theory for transport operators in LP spaces (1 < p < (0). The formalism presented in Chapter 12 is based, in a crucial way, on a characteristic property of the L1 norm, namely the additivity on the positive cone. We point out the existence of a formalism in general Banach lattices by T. Umeda [9]. However, in LP spaces (1 < p < (0), apart from the case where the velocity space is bounded away from zero, T . Umeda's assumptions are hardly compatible with natural assumptions on the cross-sections. An alternative approach is provided by factorization techniques used by many authors, particularly by T. Kato [4] and S.C. Lin [5] . We will present some of S.C. Lin's results [5] in reflexive Banach spaces and show how they can be used in the context of transport theory. As in the preceding chapter, we only deal with the existence of wave operators. Actually, S.C. Lin deals with unbounded perturbations and this makes the theory quite technical. We will restrict ourselves to bounded perturbations. This allows a considerable simplification of the different proofs and, moreover, some of S.C. Lin's assumptions are no longer necessary in this context. We consider the following framework. Let X be 291
292
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a reflexive Banach space and let {Uo(t)j t E R} be a co-group acting on X, with generator To. We denote by {U(t)j t E R} the co-group generated by T = To + B where B E L(X) . The general philosophy of the factorization technique is that the perturbation B be factorizable as B = B1B2 where B1 E L(WjX) , B2 E L(Xj W)
(for some auxiliary Banach space W) in such a way that B2 be "smooth" relative to {Uo(t)j t E R} and Bi be "smooth" relative to {UO'(t) j t E R} the dual group in the dual space X*. The concept of smoothness is explained below. As will be seen in the applications to transport theory, it is natural to allow W to be different from the initial space X . We will restrict ourselves to one wave operator, for example s lim U(t)Uo(-t). t-++oo
The other wave operators can be handled by exactly the same techniques. The smoothness assumptions we need for the analysis of this wave operator are the following. There exists p E ]1, oo[ such that, for all x E X and x* E X*,
1 0
-00
IIB2UO(s)xll~ ds < 00
r+oo
'Jo
IIB~Uo(s)x*II'. ds < 00
(13.1)
and
(13.2) where IIlIw is the norm in W, IIIIL(W) is the corresponding operator norm in L(Wj W) and p* is the conjugate exponent of p. We point out that the reflexivity assumption on X is intended to ensure that the dual group be strongly continuous. When dealing with applications to transport equations in V' spaces (1 < p < 00), we will see some interesting differences, with respect to the previous L1 theory, as regard to the relevant assumptions on the cross-sections and, in particular, the role of the velocity measure df.L(v) in the neighborhood of the origin is more transparent.
13.2
A preliminary result
We give a technical result related to Baire category theorem we need below. Let Y1 and Y2 be two Banach spaces and let H(Y1, Y2 ) := {Z: [0, oo[
--+
L(Y1' Y2 )j Z is strongly continuous} .
293
Chapter 13. Link factorization formalism For each q E [I,cm[ we define the vector space
For the reader's convenience we give a proof of the followingtechnical result. Lemma 13.1 For all Z E Hq(Yl, Y2)
Proof: Let n E N and
Clearly c, is a continuous half-norm in since {Z(t); t bounded. On the other hand, by assumption,
> 0) is locally
Thus c : Y E Yl
+
(LODllz(t)~ll;~)~
is a lower semicontinuous half-norm in Yl. Hence, for each n E N,
Mn := {y E Yl; ~ ( y I ) n) is closed and Yl = U ~ E N M ~ . It follows, from Baire's category theorem (see, for instance, [2] p. 15), that there exists an integer no such that M,, has a nonempty interior. Let D(yo,a) c Mn, be a closed ball with radius a centered at yo. Thus
The half-norm property yields sup C(Y)I a-1 YO) Ilvlly, 51
+ no] < +oo
which ends the proof. 0 We endow Hq(Yl, &) with the norm (13.3). We point out that Hq(Yl, 5) is not complete. We denote by Hq(y1, 5)its completion.
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13.3
On the wave operator s limt-++oo U(t)Uo( -t)
Yet some preliminary results are necessary. Let p E J1, 00[ . We introduce
where p* is the conjugate exponent of p and
Q: Z E Hp.(W*)
---->
1t B;Uo(t - s)B2Z(s)ds.
Clearly QZ is strongly continuous. Some properties of Q are given in Lemma 13.2 Let (13.2) be satisfied. Then
Q
E
r
CXJ
L(Hp• (W*); Hp. (W*)) and IIQII :S Jo
IIB 2 UO(s)Bl IIL(W) ds .
Denoting still by Q the unique continuous extension to H p. (W*) we have
Proof Let Z E Hp. (W*) and x· IIQZ(t)x*II' . is less than or equal to ~
t
[1 IIB;Uo (t-S)B 2 11 dS ] Let
1
W*.
By Holder's inequality,
t
11IB;uo(t-s)B21111Z(s)x*II'.ds.
CXJ
f =
Then
p
E
1
IIB;Uo (s)B211 ds.
CXJ
.I.1
CXJ
IIQZ(t)x*II'. dt :S f2(,
IIZ(s)x*II'. ds
whence
IIQZII :Sf· IIZII· This proves the first claim. To prove that QZ E Hp.(W*) even if Z E H p. (W*) we argue as follows . Let {Zn} C Hp. (W*) such that Zn ----> Z in H p. (W*) . Let t > 0 be fixed arbitrarily. Then
IIQZn(t)x· - QZrn(t)x*IIw·
:S
J:
IIBiUO'(t - S)B211 IIZn(s)x· - Zrn(s)x*IIw. ds
295
Chapter 13. Lin 's factorization formalism and
SUPtE[O,IjIIQZn(t)X· - QZm(t)x*llw·
~
(I:
IIBiUO'(s)B:iII P
ds)*(/~ IIZn(s)x* -
Zm(s)x*II'. ds)';' .
Hence {QZn(t)X* ;t E [0, 't]} nEN is a Cauchy sequence in C( [0, t] ; W*) and this shows that Q Z is strongly continuous. The following result shows that the perturbed group {U(t) ;t E R} inherits the smoothness properties of {Uo(t) ;t E R} . Lemma 13.3 Let (13.1) and (13.2) be satisfied and let rO"(Q) < 1. Then
10
00
IIBiU*(s)x*II'. ds
0,
IB2U0(t)BlcplP
Jv Jbl(x- tv1v,v1)I I~ ( x - t v , ~
5 lb2(xl~)lp llh~llt Hence
and consequently P
I I ~ ~ ~ ~ ( t ) BI l cIlh2lltpl]~
SUP
(x,.')
[/V
C(V) Ib2(x + t v l ~ ) l~P b l ( x ~ v ~IIvIIp ~)l]
Thus
Finally, using the fact that bz(z, v) = bl(z, v, v') = 0 if z
6 R,
4 and therefore E = 11 b2 1ILm 11 h2 llLm is the desired constant. In particular we have the following
0
Corollary 13.2 Let bl(., ., .) E L*(Rn x V x V) and assume that
is integrable at infinity. Then
11 B2UO(t)B111 dt < 00.
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306
Remark 13.1 When dJL(v) = dv (the Lebesgue measure) the integmbility of IIB2UO(t)Blll at infinity is ensured if n > p. To deal with the case n :S p it is necessary to impose conditions on b1 (x, v, v') at small v. We leave the details to the reader. Finally, in the case where the assumption of Corollary 13.2 holds, the estimate
indicates the mnge of the different pammeters ensuring the important conoo dition Jo IIB 2 UO(t)Blll dt < 1. In particular, this assumption is fulfilled if the diameter of n is small enough.
13.6
Comments
The material in this chapter was taken from M. Mokhtar-Kharroubi [6J. S.C. Lin's theory [5J was extended to non-reflexive Banach spaces by D.E. Evans [3J. The factorization formalism described in this chapter extends without difficulties to countable factorizations [6], i.e. to perturbations (13.16) where B2' E L(Xj wm), BI" E L(wmj X) , wm (m E N) are auxiliary Banach spaces and the series (13.16) is normally convergent, i.e. 00
L
m=l
IIBI"IIIIB2'11 < 00 .
This countable factorization could have, in principle, applications to transport equations. Indeed, if the kernel of the collision operator B, i.e. the scattering kernel, can be expanded in the form 00
b(x, v, v')
=
L
ff'(x, v)f;'(x, v)
m=l
00
L
m=l
Ilff'llllf;'11 < 00
(13.17)
Chapter 13. Lin's factorization formalism
307
wm
then the countable factorization formalism could be applicable with = LP(Rn). The condition (13 .17) expresses a kind of nuclearity ofthe collision operator B with respect to velocities. This is not, a priori, an artificial assumption since B is an integral operator with respect to velocities. However this approach is somewhat abstract since the functions fr, f2' (m E N) are unknown and it is not possible to translate, in terms of conditions on b( ., ., .), the different assumptions (besides (13.17)) we need on fi, f2' (m EN). Let us comment briefly on the assumptions (13.1) and (13.2), in the context of transport operators, say, of Section 13.5. Since b1 (., v , v') and b2 (. , v') are supported by c Rn bounded, then IIB2 UO(t)cpll and IIBiUo(t)cp*11 involve, in fact, local (in x) norms and as such they go to zero as t -> ±oo. This is the locally decaying property of free transport groups (see J. Voigt [10]). However, (13.1) imposes a sufficiently fast decaying in order to reach the integrability condition. The condition (13 .2) is yet much stronger since it involves a uniform (with respect to cp) integrability. One can interpret this better time decaying as an effect of velocity averages or, more precisely, the behaviour of the velocity measure dp,( v) at small velocities. This dispersive effect of velocity averages was already exploited, in the context of non-linear kinetic equations, by C. Bardos and P. Degond [1]. We refer to B. Perthame [7] and references therein, for recent developments on dispersive effects for kinetic equations.
n
References [1] C. Bardos and P. Degond. Global existence for Vlasov-Poisson equation in 3 space variables with small initial data. Ann. Inst Henri Poincare. Anal Non Lineaire 2 (1985) 101-118. [2] H. Brezis. Analyse Fonctionnelle: Theorie et Applications. Masson, Paris, 1983. [3] D .E. Evans. Smooth perturbations in non-reflexive Banach spaces. Math. Ann. 221 (1976) 183-194. [4] T . Kato. Wave operators and similarity for some non-self-adjoint operators. Math. Ann. 162 (1966) 258-279. [5] S.C. Lin. Wave operators and similarity for generators of semigroups in Banach spaces. Trans. Amer. Math. Soc. 139 (1969) 469-494. [6] M. Mokhtar-Kharroubi. Scattering problems in transport theory. Talk given in University of Tubingen, Germany, June 1994. Unpublished.
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Topics in Neutron Transport Theory
[7J B. Perthame. Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Part. Diff. Eq. (1996). [8J W. Greenberg, C. Van der Mee and V. Protopopescu. Boundary Value Problems in Abstract Kinetic Theory. Birkhauser Verlag, 1987. [9J T. Umeda. Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators. J. Math. Soc. Japan 38(4) (1986) 617-625. [lOJ J . Voigt. On the existence of the scattering operator for the linearized Boltzmann equation. J. Math. Anal. Appl. 58 (1977) 541-558.
Chapter 14
Inverse scattering and albedo operator. By M. Choulli and P. Stefanov
14.1
Introduction
Consider the Boltzmann equation
~~ = -v·\7xu(t,x,v)-aa(x,v)u(t,x,v)+
i
k(X,VI,V)U(t,x,VI)dv' (14.1)
in Rn x V 3 (x, v), V being an open subset of Rn , n 2: 2. Equation (14.1) describes the dynamics of a flow of particles in R n under the assumption that the interaction between them is negligible (no non-linear terms). This is the case for example for a low-density flow of neutrons. The term involving aa describes the loss of particles from (x, v) ERn x V due to absorption or scattering into another point (x, Vi), while the last term in (14.1) involving k represents the production at x E Rn of particles with velocity v form particles with velocity Vi . The total rate of this production at (x, Vi) is given by
ap(x, Vi) =
i
k(x,v',v)dv.
Following [14] we say that the pair (aa, k) is admissible, if (i) O:S aa E Loo(R n x V), (ii) O:S k(x, Vi,.) E Ll(V) a.e. (x, Vi) ERn x V and a p E Loo(Rn x V)
309
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310
(iii) There is an open bounded set X eRn, such that k(x,v',v) and
Ga(x, v) vanish if x
f/. x.
Denote To = -v · \7 x with domain D(To) = {J E p(Rn x V); V · \7 x f E LI(Rn x V)} . It is well-known that To is a generator of a strongly continuous group Uo(t)f = f(x - tv, v) of isometries on LI(Rn x V) preserving the non-negative functions. Following the widely accepted notations, let us introduce the operators
-Ga(x, v)f(x, v), [A 2f](x, v)
=
J
v k(x, v', v)f(x, v') dv' , T = To + Al
+ A2 =
TI
+ A2
and set A = Al + A 2. Operators Al and A2 are bounded on P (R n x V) and T I , T are generators of strongly continuous groups UI(t) = etT1 , U(t) = etT , respectively [14]. For UI(t) we have an explicit formula
[UI(t)f](x , v) = e -
Jo'
C7
a
(x-sv,v)ds
f(x - tv, v),
(14.2)
while for U (t) we have (14.3) We work in the Banach space LI(Rn x V), so here IIU(t)11 is the operator norm of U(t) in £(p(Rn x V)). It should be mentioned also that U(t) preserves the cone of non-negative functions for t :::: O. One can define the wave operators associated with T, To by s-lim U(t)Uo(-t),
(14.4)
s-lim Uo( -t)U(t) .
(14.5)
t-+cx>
t-+cx>
If W _,
W+ exist,
then one can define the scattering operator
S=W+W_ as a bounded operator in LI(Rn x V). Scattering theory for (14.1) has been developed in [8], [15], [21] and we refer to these papers (see also [14]) for sufficient conditions guaranteeing the existence of S . We would like to mention here also [11], [19], [6], [16], [22] . An abstract approach based on the Limiting Absorption Principle has been proposed in [10]. We will show in Section 14.2 however that S can always be defined as an operator S : L~(Rn x V\ {O}) -> L~oc(Rn x V\ {O}). The first inverse problem we are interested in is the following: Does S determine uniquely G a , k? We show that the answer is affirmative if G a is independent of v.
Chapter 14. Inverse scattering and albedo operator
311
Theorem 14.1 Let (O'a, k), (u a , k) be two admissible pairs such that O'a, u a do not depend on v and denote by S, S the corresponding scattering operators. Then, if S = S, we have O'a = Ua , k = k.
One can relax a little bit the assumption that O'a, u a do not depend on v . For example, assume that V is spherically symmetric and O'a = O'a(x, Ivl), u a = ua(x, Ivl). Then it is clear from the proof (see also (14.6) below) that the uniqueness result in Theorem 14.1 still holds. However, it is important to note that Theorem 14.1 fails to be true for general O'a. Consider for example the pairs (O'a, 0), (ua , O) , where u a = O'a(x + p(x, v)v, v), with p some nontrivial continuous function such that p(x, v)v is bounded on Rn x V . Then, if (O'a , 0) is admissible, so is (u a , 0). Since k = 0, we have Sf = e -
J::= ua,{x-sv,v)ds f,
(k = 0)
(14.6)
and it is easy to see that S = S although O'a =I- u a. Note that if k = 0, and O'a does not depend on v, it follows from (14.6) that S determines uniquely the X-ray transform of O'a and therefore O'a. The proof of Theorem 14.1 is constructive, it implies an explicit procedure for recovering O'a and k from S. It turns out that all the information necessary to recover 0'a, k is contained in the behavior of the Schwartz kernel S(x,v,x' , v') of S near the singularities (x,v) = (x' , v') and x = x', v =I- v', respectively. Next object we consider is the so-called albedo operator $. Assume that X is convex and has CI-smooth boundary ax. We propose the following definition of $ which generalizes that given in [1], [7], [12]. Denote f ± = {(x , v) E ax x V; ±n(x) · v > O} , where n(x) is the outer normal to ax at x E ax. Consider the measure d1, = In(x) . v ldJ.£( x)dv on f ±, where dJ.£(x) is the measure on ax. Let us solve the problem
(at - T)u
=
0 in R x X x V,
Ult«o
=
0
(14.7)
for u(t,x , v) , where g E L~(R; £l(f_, d~)) and T is considered as a differential operator in X x V. We will see that (14.7) has a unique solution u E C(R; LI(X x V)) and one can define the albedo operator $ by (14.8)
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312
Operator $ : L~ (R; L1 (r _, d~)) - t L{oc (R; L1 (r +, dO) maps the incoming flux on ax to the outgoing flux on ax. It can be seen that $g can be defined more generally for 9 E L1 (R x r _, dt d~) with 9 = 0 for t «0. It has been shown in [1], [7], [12] that there is a relationship between Sand $. We show below that in fact $ determines S uniquely and conversely, S determines $ uniquely by means of explicit formulae. To this end, let us define the extension operators E± and the restriction (trace) operators R± as follows. Set D = {(x, v) ERn x V\ {O}; ::It E R, such that x - tv EX},
(14.9)
and define the functions T±(X, v)
= max{t E R;
x ± tv E aX},
(x, v) ED.
Given 9 E L1 (R x r ±, dt d~), consider the following operators of extension: g(±T±(X,v), x ± T±(X, v)v, v),
(E±9) (x, v)
=
(x, v) E D
{ 0, otherwise.
It is easy to check that E± : L1 (R x r ±, dt~) Denote by R± the operator of restriction
-t
£1 (R n x V) are isometric.
Although R± is not a bounded operator on £1(Rn x V) (see [2], [3]), R±Uo(t)f E L1(R x r ±, dt~) is well defined for any f E L1(Rn x V) (see (14.41)). Denote by Xf! the characteristic function of D. We establish the following relationships between Sand $. Theorem 14.2 Assume that X is convex. Then (a) $g = R+Uo(t)SE_g, 9 E L~(R x r _,dt~), (b) Sf = E+$R_Uo(t)f + (1- Xf!)f, f E L~(Rn x V\{O}), (c) $ extends to a bounded operator
~f and only if S extends to a bounded operator on L1(Rn x V).
Remark 14.1 Let us decompose L1(Rn x V) = L1(D)EB£1((Rn x V)\D) . A similar decomposition of course holds for L~(Rn x V\ {O}). Then S leaves invariant both spaces, moreover SIL'((Rn XV)\f!) = Id, so S can be decomposed as a direct sum S = S1 EBI d . Denote R± = R±Uo( . ) : £1 (D) - t
Chapter 14. Inverse scattering and albedo operator
313
Ll(R x r ±, dtdO. We will see in Section 14.4 that R± are isometric and invertible and Rj/ = £± with £± : Ll(R x r ±, dtd!,) - t Ll(O) , £±f := E±flu(!1). Then we can rewrite Theorem 14.2 (a) , (b) in the following way
or even more simply as
with SI = SIL~(!1) as above. Thus $ can be obtained from SI by a conjugation with invertible isometric operators and vice-versa. Remark 14.2 The albedo operator is defined in [lJ in somewhat different manner by (14 .8), provided that u solves (14.7) for t > 0 and u satisfies Ult=o = 0 instead of Ult«o = 0 (in fact, it is assumed that u satisfies a non-zero initial condition, but this can be easily reduced to the case of zero initial condition). The relationship between $ and S established in [1] can be written as R+SUo(t) = $R-Uo(t), which can be obtained as a consequence of Theorem 14.2(b) (or (a)) . Remark 14.3 Some of notations above are a little bit ambiguous. Namely, the expression E+$R_Uo(t)f in Theorem 14 .2(b) seems to depend on t, while the left-hand side of the equality in which it is involved is independent oft. In fact, here Uo(t)f is a function of x, v and t is a parameter, the same applies to R-Uo(t)f . Since the operator $ acts on functions g = g(t, x, v) depending on t as well, we consider now t as a variable and apply $ to the function (t, x, v) f-+ R-Uo(t)f . The result is a function oft, x and v. Next we apply E_ and obtain a function of x and v only. Perhaps a more precise notation in Theorem 14.2(b) would be E+$R_Uo( )f .
An immediate consequence of Theorem 14.2 is that $ determines uniquely k for (J a independent of v and X convex. However, we can prove this for not necessarily convex domains as well independently of Theorem 14.2. (J a,
Theorem 14.3 Let ((Ja , k), (aa, k) be two admissible pairs with (Ja, aa independent of v and denote by X any open bounded set with Cl-smooth boundary with the p;:.operty that (J a, k, aa, k vanish outside X . Then, if the albedo operators $, $ on ax coincide, we have (Ja = a;; , k = k .
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It should be mentioned that in the case where k is of the form k
=
RK + Q with R, Q known and K = K(x, v), the uniqueness of the inverse problem studied in Theorem 14.3 has been investigated in [13] for convex X
under some smallness assumptions that guarantee that the corresponding integral equations can be solved by successful approximations. We would like to mention here also [5], where the problem of determination of k from the stationary albedo operator in the one-dimensional case is considered. The proof of Theorem 14.3 is constructive as well. We study the Schwartz kernel of $ and describe the first two singular terms in it. We show that a a can be recovered from the first term, while the second one determines k, similarly to the proof of Theorem 14.l. Finally, we would like to mention that we have found some analogy between the albedo operator $ and the Dirichlet-to-Neumann map A related to the boundary value problem for the Schrodinger operator or for the conductivity operator \1 . 'Y\1, 'Y = 'Y(x) > 0 [17]. More precisely, denote by A the operator acting on the boundary of a bounded domain mapping the Dirichlet data of the solution to (-Ll + q)v = 0 (respectively \1 . 'Y\1v = 0) to its Neumann data. As proven in [17], A determines uniquely q (respectively 'Y). We found that $ in our case is in some sense an analogue to A or more precisely to the time-dependent Dirichlet-to-Neumann map associated with the wave equation (8;- Llx + q)v = o. It is well-known that there is a close relation between the scattering operator for the Schrodinger equation and A. Theorem 14.2 we proved can be considered as an analogue of this result in transport theory. We would like to mention however, that the Schrodinger equation and the Boltzmann equation have quite different properties. The material in this chapter is taken from [4]. It should be mentioned that the main theorems can be generalized to the case where aa, k depend on t as well.
14.2
The special solution and the scattering operator
An important role in our analysis is played by the following special solutions. Given (x', v') ERn x V\ {O} consider the following problem
8(x - x' - tv)8(v - v'),
(14.10)
Chapter 14. Inverse scattering and albedo operator
315
8 being the Dirac delta function. We will show that (14.10) has unique solution u(t, x, v, x', Vi), with u depending continuously on t with values in V' (R~ x Vv x R~, X Vv' \ {O} ). Moreover, we have the following singular expansion of U. Theorem 14.4 Problem (14 .10) has unique solution where
Uo
= e - Joroo CT,,(x-sv,v)ds uI:( x
- x1 -
_ Jex> e- Jor' CT,,(x-7'v,v)d7' e -
ul =
t) v uI:( V -
u = uo + UI + U2, V')
roo CT,,(X-SV-7'v',v')d7'
Jo
o xk(x - sv, Vi, v)8(x - sv - (t - s)v' - x') ds and
Proof: Pick 'P E
C~(Rn
x V\ {O}) and consider the problem
Wlt«o
=
'P(x - tv, v),
(14.11)
Since min{lvl; (x, v) E 'P for some x} > 0, there exists to = to('P), such that Uo(t)'P = 0 for x E X, t < -to. Then w := U(t
+ to)Uo( -to)'P
(14.12)
solves (14.11) and it is easy to see that w does not depend on the particular choice of to. Applying Duhamel's principle (14.13) we get
Applying Duhamel's formula once more, we obtain
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316
where
Wo = U1(t WI
=
W2 =
+ to)UO( -to) S2 + to and for S2 < -to (provided that Sl > 0). Therefore, the following integral is well-defined
U2 :=
jt
roo U(t _ s2)M(Sl, S2,
-00
Jo
, ,x', v') dS 1 ds 2,
and we have
U2 E C (Rt; L~c(Rn x, x Vv' \ {O}, L1(Rn x x Vv ))) . On the other hand, by (14.16)
W2(t, x, v)
=
r
JRnxV
U2(t, x, v, x', V/)cp(X', v') dx ' dv' .
(14.20)
We are now ready to conclude the proof of Theorem 14.4. We found (see (14.14), (14.15), (14.20)) that the unique solution to (14.11) has the form w = (u(t,x,v, , . ),cp), where u = Uo + U1 + U2 is a distribution with properties as stated above. It is clear now that u solves the transport equation in distributional sense and satisfies the initial condition in (14.10) as well, therefore u solves (14.10). Moreover, this solution is unique because the solution to (14.11) is unique. 0 We will prove next that the wave operators W _, tv+ always exist as operators between suitably chosen spaces. Proposition 14.1 The limits W_, W+ exist as operators between the spaces
W_ : L~(Rn x V\ {O}) ~ L1(Rn x V) W+: L1(Rn x V) ~ Ltoc(Rn x V\ {O}).
Chapter 14. Inverse scattering and albedo operator
319
Proof: Pick fJ(RnxV\{O}) . Sincemin{lvl i (x,v) Ef for some x} > 0, for some to = to(f) we have Uo(-t)f = 0 in X for t > to. Moreover, U(t)Uo(-t)f = U(to)Uo(-to)f for t ;:::: to and therefore W-f = U(to)Uo(-to)f. This proves in particular that the limit W_ : L~(Rn x V \ {O}) -+ L I (Rn x V) (see (14.1» exists as an operator between these two spaces. Next, let us fucg E LI(Rn x V) and a compact set K c Rn x V\{O} and consider [Uo( -t)U(t)g](x , v) for large t and (x, v) E K. We claim that this is independent of t for t > tl with some tl = tl(K). In particular, this would prove that the limit W+ (see (14.5» exists as an operator W+ : LI(Rn x V) --+ Lfoc(Rn x V\ {O}) and W+gIK = Uo( -tl)U(tl)gIK. Indeed, the Duhamel's principle U(t) = Uo(t)
+
lot Uo(t - s)AU(s) ds
(14.21)
implies
Uo(-t)U(t)g = 9 +
lot Uo(-s)AU(s)gds .
(14.22)
Since AU(s)g = 0 for x fJ. X, we have Uo(-s)AU(s)g = (AU( s)g)(x + sv, v) = 0 for (x, v) E K , s > tl = tl(K) . Therefore, Uo( -t)U(t)9IK does not depend on t for t > tl and our claim is proved. 0 depending on x and X, and we denoted W = { v ; 3x, such that ( x ,v ) E suppx). The last integral is taken over the bounded set
DE= { ( x , v l ,v ) , x E X ,v E W , Iv - v'l
< E).
Chapter 14. Inverse scattering and albedo operator
323
Let us estimate the measure meas(D,). We have meas(D,) =
J
W
11 X
dv' dx du = Cen (u-v'(<E
JW
dx dv = C'E".
Therefore, in (14.26) we have a locally integrable function (see (ii), (iii) in Section 14.1) and the integration is performed over a set D, with meas(D,) + 0. Since the Lebesgue integral is absolutely continuous with respect to the Lebesgue measure, we get that
in L1 (Rnx V). Finally, we have
JSIJJ 5
S2(x, v, x', vl)&(x, v, XI, vl) dx' dv' dx dv
lS2(x,v,x',u')l dxdvdx'dv' Ec
with E, = {(x, v, x', v'); (x, v) E supp X , lx - x'l 5 E, Iv - v'l 5 E). There exists EO > 0 such that for 0 < E < EO we have E, c E,, C Rn x V\{O) x R n x V\{O) and S2is an integrable function on E,, by Theorem 14.5. As before, it is easy to see that meas(E,) = O(E'~)+ 0 as E + 0. Therefore, (14.28) tends to 0 and we obtain
in L1(Rn x V). Now, (14.25), (14.27) and (14.29) together complete the proof of Proposition 14.5. 0 Assume now that a, is independent of v. We deduce from Proposition 14.5 that one can recover
for a.e. (x, v) E Rn x V. Since V is an open set, we see that we know (14.30) for a.e. (x, v) E Rn x U,where U is a small neighborhood of some vo E V\{O). Thus we can recover the X-ray transform
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324
of l7a (x) for a.e. (x, w) E Rn x U, where U is a small neighborhood of vo/lvol in sn-l The latter is sufficient to recover l7a (see e.g. [9]) . We note that in the particular case where for any w E sn-l, the velocity space V contains some v of the kind v = rw with r = r(w) > 0, we can recover the X-ray transform (14.31) for a.e. (x,w) ERn x sn-l and therefore we can write an explicit formula [9] for l7a (x). We proceed next with the reconstruction of k(x, Vi, v). Choose two functions cP E C.;"'(Rn) with ~ cP ~ 1, cp(o) = 1, cp(x) = for Ixl > 1 and CPl E C.;"'(R) with CPl(s)ds = 1, ~ CPl· Set W = {(v',V) E V x Vi v =J. 0, Vi =J. 0, v =J. Vi} . For 101> 0, 102> set 'l/Jc l,c2 equal to
J
xcP ( -1 ( X 102
-
X
I
-
°
°
°°
(x-xl).(V-V Iv - v'12
I)(
V -
V
I)))
.
(14.32)
Proposition 14.6 With 'l/Jc l,c2 as above we have
O
10 I
'PI(-~) ds ci
E(O, x, v', v)k(x, v', v).
(14.34)
Finally, choose a compact set K eRn x Wand consider
I IS2(X, v, x', V')'ljJcl.C2 (x, v, x', v') dx'i dxdv'dv K
0 we have
+
Proof: Given f E L1(Rn x V), set f = fl f2 with fl = Xnf, f2 = (1 -xn) f . Using Duhamel's principle (14.13), we see that U(t) f2 = Uo(t)f2 and thus by (14.41), &U(t) f2 = 0. For fl we have by using (14.21) and
Topics in Neutron Transport Theory
328 (14.41)
f:a IIR±U(t)fI IILl(r±.de)dt
:S: IlfIIILl(RnxV) + fa 11ft R±Uo(t - s)AU(s)fI dsll -a
0
dt £l(r±.~)
:S: IlfIIILl(RnxV) + f:a f:a IIR±Uo(t -
s)AU(s)fIIILl(r±.~)dsdt
IlfIIILl(RnxV) + f:aJ:a IIR±Uo(t -
s)AU(s)fIIILl(r±.~)dtds
=
:S: IlfIIILl(RnxV) + J:a IIUo( -s)AU(s)fdILl(Rnxv)dS :S: (1 + 2allo-pllLooeallupllLOO) IlfIIILl(RnxV)' Lemma 14.2 R-U(t)fIR+xr _
=
R-Uo(t)fl~xr _
for any f E Ll(Rn x V).
Proof' We have to show that
By inspecting the proof of Lemma 14.1 we see that it suffices to prove that
which is equivalent to
In order to complete the proof, it is enough to observe, that
R_Uo(t)hI R+xr _ = 0 for any h with h(x, v) = 0 for x ¢. X. Given g E L~(RiLl(r_,d~)), consider the problem (14.7) Proposition 14.7 Problem (14.7) has a unique solution in C(RiLl(X x V)) given by u = U(t)W_E_glxxv,
Chapter 14. Inverse scattering and albedo operator
329
Proof: Note first that the uniqueness follows from the fact that the homogeneous problem (with g = 0) has only a trivial solution, because the transport operator with boundary conditions ulRxr- = 0 generates a continuous semigroup of solution operators. Next, note that if to is such that g = 0 for t < -to, we have Uo(t)E-g = 0 in X x V for t < -to and moreover, U(t)Uo(-t) E-g = U(to)Uo(-to)E-g for t > to, so although E-g does not necessarily belong to L,!(Rn x V\{O)) (see (14.1) and Proposition 14.1), the limit W-E-g trivially exists. Set w = U(t)W-E-g = U(t + to)Uo(-to)E-g. Then w clearly solves the Boltzmann equation in R x X x V. We have that R-wlt,-to = 0 and by Lemma 14.2 and (14.38), R-wit>-to = R-Uo(t)E-glt>-t, = glt>-to = g. Therefore, w satisfies the boundary condition as well. Thus setting u = wlxxv, we get a solution to (14.37). 0 We see now that the definition of $, given in (14.8) $g = R+U, $ : L:(R; ~ l ( r - ,cy)) +L;,,(R;
~ l ( r +cy)), ,
(14.42)
u being the solution to (14.7), is correct. Indeed, by Proposition 14.7, $g = R+U(t) W-E-g and by Lemma 14.1, $g E L;,,(R; L1(I'+, @)). We note that in fact, $g is well-defined also for g E L1(R x I?-, d t e ) with g = 0 for t O. Clearly, meas(E",) -> 0, as c -> 0, where meas(E",) is associated with dTdE,' dE,. On the other hand, by Theorem 14.6 the integrand in the last integral is a £I-function. AB before, we conclude from this that the limit in (14.65) is zero, as stated. Combining (14.64), (14.65) we complete the proof. By Proposition 14.10 one can recover the X-ray transform of CTa(X), provided that CTa is independent of v and therefore one can recover CTa itself. We proceed with the recovery of k. Next proposition is an analogue of Proposition 14.6. Let '1/J"'l' ' 2 and W be the same as in Proposition 14.6.
338
Topics in Neutron Transport Theory
Proposition 14.11 We have
lim lim G+(O)!! a(t - t/, X, v, x', v') Rax
0' -->0 02-->0
=e -
fc
/ T- ( Z , V
0
)
(
1
')
O"a X-PV ,v dP
') k ( x,v,v
(14.66)
where the integral is to be considered in distribution sense and the limit holds in LfoAX). Remark 14.4 The restriction of'ljJo, ,02 (x-tv , v , x'_t'v / , v') on R t x r + x Rt' x r _ \ {v = v'} is not necessarily a function of compact support on that variety, but as will be seen from the proof of Proposition 14.11, the formal integral above is well defined. Operator G+(O) above (see (14 .50)) is applied to the formal integral considered as a function oft, x, v. Proof: Note first that for v =I- v' we have ao = O. Next, for al we get
!! ! E(s,x,v/,v)k(x-sV,V/,V)B(x-sv, X)ol,CPl(t~S)ds,
al (t - t/, x, v, x', v') 'ljJo,,02 (x - tv, v, x' - t'v / , v/)dp,(X/)dt'
R
ax
=
f' ( x-sv ,v' ) ) = e - J,f80 O"a(x-pv ,v)dp e- o J, O"a (x- s v-pv ,v )dp Fun were h . cE( s, x, v',v tion s --+ E(s,x , v/,v)k(x - sv,v/,v)B(x - sv,x) is integrable with values in Ll(r+ x VV/ d1, dv/). Therefore, as Cl --+ 0, the limit above exists in Ll (Rs x r + x VV/, ds d1, dv / ) and we have I
I
lim lim !!al(t-t/,x,v,x/,v / ) Rax
0' -->002-->0
E(t , x, v', v)k(x - tv, v', v)B(x - tv, x).
(14.67)
By applying G+(O) to both sides of the equality above we get that (14.66) holds with a = al'
Chapter 14. Inverse scattering and albedo operator
339
Finally, let us fix a compact set K c r + x Vv ' that does not intersect the varieties v = v', v' = O. Then for any a > 0 we have a
JJ JJ
l l a2(t-t ,x,v,x ,v' )
-a K
R
ax
JJIn(xl)·v/l-lla2(t-tl,x,v,xl,vl)ldt/~~/dt, a
