Critical spectrum and spectral mapping theorems in transport theory Mustapha Mokhtar-Kharroubi and Mohammed Sbihi Laboratoire de Math´ematiques de Besan¸con. Universit´e de Franche Comt´e. 16, Route de Gray, 25030 Besan¸con. France. E-mail : [email protected]; [email protected]

Abstract We give spectral mapping theorems for general neutron transport semigroups on unbounded convex domains. The mathematical analysis relies on the critical spectrum [12], some related stability results [3] and Fourier analysis of transport equations.

1

Introduction

Let Ω ⊂ RN be an open set and let µ be a positive Radon measure on RN with support V. The streaming semigroup is defined by Z t − σ(x − sv, v)ds U (t) : Lp (Ω × V ) 3 ϕ 7→ e 0 ϕ(x − tv, v)χ{t 0; (x − sv) ∈ / Ω} and σ ∈ L∞ (Ω × V ; dx ⊗ dµ(v)) is the collision frequency. We denote by T the generator of (U (t))t≥0 . Let Z p K : L (Ω × V ) 3 ϕ 7→ k(x, v, v 0 )ϕ(x, v 0 )dµ(v 0 ) V

be the collision operator with scattering kernel k(x, v, v 0 ). If K is bounded, then T + K generates a C0 −semigroup (V (t))t≥0 called the transport semigroup and is given by a 1

Dyson-Phillips expansion V (t) =

∞ X

Uj (t)

(1)

0

where

Zt U0 (t) = U (t), Uj+1 (t) =

U0 (t − s)KUj (s)ds (j ≥ 0). 0

Since the pioneering papers by J. Lehner and M. Wing [5][6], it is customary to consider time dependent neutron transport equations as Cauchy problems ½ dϕ dt = T ϕ + Kϕ (2) ϕ(0) = ϕ0 . For a long time, the time asymptotic behavior (t → ∞) of such Cauchy problems was dealt with according to the following pattern : 1) Analysis of the spectrum of T + K. Typically, in a bounded domain Ω, some power of (λ − T )−1 K is compact and σ(T + K) consists of a half plane (the spectrum of T ) {λ; Reλ ≤ −λ∗ } and, at most, isolated eigenvalues {λi }i∈I , (Reλi > −λ∗ ), with finite algebraic multiplicities where Zt 1 ∗ λ = lim inf σ(x − sv, v)ds (3) t→∞ {(x,v);t≤τ (x,v)} t 0

−λ∗ .

and {λi ; Reλi ≥ α} is finite for all α > 2) Express the solution V (t)ϕ0 of (2) as an inverse Laplace transform of the resolvent (λ − T − K)−1 . Then, by deforming the path of integration and picking up the residues corresponding to the poles (i.e. the eigenvalues of T + K), an asymptotic expansion of V (t)ϕ0 in terms of finitely many eigenvalues is derived. This approach imposes some (unnatural) constraints on the initial data, typically ϕ0 ∈ D((T +K)2 ) (see [4][14] and references therein for recent developments in this direction). It was I. Vidav [15] who first pointed out that even if σ(T +K)∩{Reλ > −λ∗ } is composed of isolated eigenvalues, n o ∗ σ(et(T +K) ) ∩ µ; |µ| > e−λ t may contain some continuous spectrum because of the lack, in general, of a spectral mapping theorem for the exponential function C 3 λ 7→ eλt . He observed that the time asymptotic behavior of {V (t); t ≥ 0} relies on its spectral analysis and showed that the compactness of some remainder term Rn (t) =

∞ X j=n

2

Uj (t)

is the relevant tool to exclude this possible continuous spectrum and restore the (partial) spectral mapping property n o ∗ ∗ σ(et(T +K) ) ∩ µ; |µ| > e−λ t = et(σ(T +K)∩{λ; Reλ>−λ }) . (4) These abstract ideas were deepened by J. Voigt [16][17], L. Weis [18], G. Schluchtermann [13] and M. Mokhtar-Kharroubi [7] [8, Chap 2]. The present paper deals with spectral mapping theorems in unbounded domains Ω where the above compactness assumptions are not satisfied. To our knowledge, this is the first work on this topic. In [7], the author conjectured the (partial) spectral mapping theorem (4) for some abstract perturbed Cauchy problems in Banach lattices of the form (2). This conjecture was proved by F. Andreu, J. Martinez and J. M. Mazon [1] under the assumption that KU (t1 )K...KU (tn )K depends continuously in the norm operator topology on (t1 , ..., tn ) (ti > 0, 1 ≤ i ≤ n) for some integer n. A more precise conjecture, for abstract perturbed Cauchy problems in Banach spaces, was given in [8, Chap 2]. This conjecture was proved by S. Brendle, R. Nagel and J. Poland [3]. In their proof an important role is played by the concept of critical spectrum introduced by R. Nagel and J. Poland [12]. In the present paper, we show how the abstract results in [3] apply to transport equations in unbounded convex domains Ω. We deal here with the case 1 < p < ∞ only. The L1 theory relies on different technicalities [11]. Our paper is organized as follows : in Section 2, we recall the properties of the critical spectrum [12] and some related stability results [3]. Section 3 is devoted to the critical spectrum of streaming semigroups (U (t))t≥0 . In Section 4, we determine the critical spectrum of transport semigroups (V (t))t≥0 and prove a partial spectral mapping theorem of the form (4) under an optimal assumption on the velocity measure µ when the scattering kernel depends ”smoothly” on the spatial variable. In Section 5, we remove this smooth dependence at the price of a stronger assumption on the measure µ and prove a full spectral mapping theorem σ(et(T +K) )\ {0} = etσ(T +K) . Our assumptions on the velocity measure µ are very general and cover in particular the usual continuous or multigroup models. We end up the paper by some conjectures. Besides the stability results given in [3], the mathematical analysis relies on interpolation and on Fourier analysis of the Dyson-Phillips expansion (1). We assume throughout this paper that the collision frequency is space homogeneous, i.e. σ(x, v) = σ(v). The authors are indebted to the referee for helpful remarks and suggestions.

2

Critical spectrum of C0 -semigroups

In this section, we recall the concept of critical spectrum introduced by R. Nagel and J. Poland [12] and some related stability results. Let X be a Banach space and τ = (U (t))t≥0 be a strongly continuous semigroup on X. We consider the Banach space

3

e := `∞ (X) of all bounded sequences in X endowed with the norm X ke xk = sup kxn k n∈N

e where x e = ³(xn )n∈N ´ . We extend the semigroup (U (t))t≥0 to X and obtain a new semie (t) group τe = U defined by t≥0

e (t)e U x := (U (t)xn )n∈N for x e = (xn )n∈N . eτ be the subspace of strong continuity of τe Let X ½ ¾ ° ° eτ := x e lim°U e (h)e X e ∈ X; x−x e° = 0 . h↓0

³ ´ e (t) b := X/ e X eτ , -invariant. On the quotient space X This subspace is closed and U t≥0 ³ ´ ³ ´ e (t) b (t) the semigroup U induces a quotient semigroup τb = U given by t≥0

t≥0

b (t)b e (t)e eτ for x eτ . U x=U x+X b=x e+X The critical spectrum of U (t) is then defined as ˆ (t)) σcrit (U (t)) := σ(U while its critical spectral radius is defined as b (t)). rcrit (U (t)) := r(U Moreover, the critical growth bound is defined as b (·)) ωcrit (U (·)) := ω0 (U where ω0 is the usual growth bound (type). We state now the main properties of the critical spectrum. Theorem 1 [12]. Let (U (t))t≥0 be a strongly continuous semigroup on a Banach space X with generator T . Then : (a) σcrit (U (t)) ⊂ σ(U (t)), (b) rcrit (U (t)) = eωcrit (U (·))t , (c) σ(U (t))\ {0} = etσ(T ) ∪σcrit (U (t))\ {0} , (d) ω0 (U (·)) = max {s(T ), ωcrit (U (·))} .

4

The critical spectrum enjoys nice perturbation properties [3]. Indeed, let K be a T −bounded operator on X such that Zh kKU (s)xk ds ≤ q(h) kxk for each x ∈ D(T ) and h ≥ 0

(H)

0

where q : R+ → R+ satisfies limq(t) = 0. Then T + K generates a strongly continuous t↓0

semigroup (V (t))t≥0 given by a Dyson-Phillips expansion (1). Assumption (H) is of course satisfied by a bounded perturbation K. Theorem 2 [3]. Let (U (t))t≥0 be a C0 -semigroup with generator T and let (V (t))t≥0 be the C0 -semigroup generated by T + K where K satisfies (H). If for some t0 ≥ 0 the mapping t 7→ R1 (t) is norm (right) continuous for t ≥ t0 , then σcrit (V (t)) = σcrit (U (t))

(t ≥ t0 ).

Theorem 3 [3]. Let (U (t))t≥0 be a C0 -semigroup with generator T and let (V (t))t≥0 be the C0 -semigroup generated by T + K where K satisfies (H). If for some k ∈ N the mapping t 7→ Rk (t) is norm (right) continuous for t ≥ 0, then ωcrit (V (·)) = ωcrit (U (·)).

3

Critical spectrum of streaming semigroups

We recall that the approximate spectrum σap (T ) of a closed densely defined linear operator T in a Banach space X is defined by σap (T ) = {λ ∈ C; ∃(xn )n ⊂ D(T ), kxn k = 1, kT xn − λxn k → 0 as n → ∞} , and σap (T ) ⊂ σ(T ) is closed. We recall some results. Theorem 4 [2]. Let (U (t))t≥0 be a C0 -semigroup with generator T. Let (λn )n ⊂ σap (T ) be such that lim |Imλn | = ∞ and lim etλn = µ. Then µ ∈ σcrit (U (t)). n→∞

n→∞

The second result concerns the invariance of the spectrum of streaming semigroups by rotation.

5

Theorem 5 [17]. Let µ {0} = 0 and α : Ω × V 3 (x, v) 7→ α(x, v) = η ∈ R, Mη : Lp (Ω × V ) 3 f 7→ e−iηα f ∈ Lp (Ω × V )

x.v . |v|2

Then, for

is an isometric isomorphism such that Mη−1 U (t)Mη = e−iηt U (t) and Mη−1 T Mη = T + iηI. As a consequence, σ(U (t)) = σ(U (t))·Γ and σ(T ) = σ(T )+iR, where Γ = {z; |z| = 1} . Similarly σap (U (t)) = σap (U (t)) · Γ and σap (T ) = σap (T ) + iR. Remark 1 It follows from Theorems 4 and 5 that etσap (T ) ⊂ σcrit (U (t)) for streaming semigroups. Before giving the main result of this section, we recall the description of the spectrum of streaming semigroups. Theorem 6 [9]. Let 1 ≤ p < ∞, and Ω be a convex domain in RN . 1. If Ω & RN and if the hyperplanes have zero µ-measure, then n o ∗ σ(T ) = σap (T ) = {λ; Reλ ≤ −λ∗ } and σ(U (t)) = σap (U (t)) = µ; |µ| ≤ e−tλ . 2. If Ω = RN and if the essential range of σ has at most countably many connected components, then σ(T ) consists of a set of disjoint slabs [ σ(T ) = σap (T ) = Λi i∈I

of the form Λi = {λ; ai ≤ Reλ ≤ bi } (ai ≤ bi ), where [ai , bi ] (i ∈ I) are the connected components of the essential range of −σ. The spectrum of the streaming semigroups is o [n σ(U (t)) = σap (U (t)) = µ; etai ≤ |µ| ≤ etbi . i∈I

Remark 2 In the above theorem λ∗ is given by (3). For the case Ω = RN , λ∗ = ess inf σ(v). We note also that sup bi = −λ∗ and inf ai = −λ∗∗ where λ∗∗ = ess sup σ(v). i∈I

i∈I

From Remark 1 and Theorem 6 we deduce : Theorem 7 We assume that Ω ⊂ RN is convex and the hyperplanes have zero µmeasure. Then σcrit (U (t)) = σ(U (t)).

6

4

A partial spectral mapping theorem

In neutron transport theory the collision operator has the form Z p k(x, v, v 0 )ϕ(x, v 0 )dµ(v 0 ). L (Ω × V ) 3 ϕ 7→ Kϕ = V

Thus, we may naturally regard K as an operator valued mapping Ω 3 x 7→ K(x) ∈ L(Lp (V )), where

Z K(x) : Lp (V ) 3 ϕ 7→

k(x, v, v 0 )ϕ(v 0 )dµ(v 0 ) ∈ Lp (V ). V

We assume that Ω 3 x 7→ K(x)ψ ∈ Lp (V ) is measurable for every ψ ∈ Lp (V ) and Ω 3 x 7→ kK(x)kL(Lp (V )) is essentially bounded. Then we define a collision operator by K : Lp (Ω × V ) 3 ϕ 7→ K(x)ϕ(x), where we make the identification Lp (Ω × V ) := Lp (Ω; Lp (V )). It follows (see [10]) that K ∈ L(Lp (Ω × V )) and kKkL(Lp (Ω×V )) = esssup kK(x)kL(Lp (V )) .

(5)

x∈Ω

We give the definition of a regular collision operator. Definition 1 [10]. A collision operator K is said to be regular if (i) {K(x); x ∈ Ω} is a set of collectively compact operators on Lp (V ), i.e. {K(x)ψ; x ∈ Ω, kψkLp (V ) ≤ 1} is relatively compact in Lp (V ). 0

0

(ii) For every ψ 0 ∈ Lp (V ), {K 0 (x)ψ 0 ; x ∈ Ω} is relatively compact in Lp (V ). We now recall two preliminary results.

7

Lemma 1 Let K be a regular collision operator. Then K can be approximated in operator norm by a sequence of collision operators with separable kernels X k(x, v, v 0 ) = αi (x)fi (v)gi (v 0 ), i∈I p0

with fi ∈ Lp (V ), gi ∈ L (V )

( p1

+

1 p0

= 1) and αi ∈ L∞ (Ω) (I finite). Moreover, if

Ω 3 x 7→ hK(x)ϕ, ψi is uniformly continuous

(6)

p0

for every (ϕ, ψ) ∈ Lp (V ) × L (V ) where h·, ·i denotes the duality pairing between Lp (V ) 0 and Lp (V ), then αi can be choosen uniformly continuous. Proof : Apart from the uniform continuity of αi , the proof is given in [10]. However, to prove this additional property we need to resume the proof of the whole theorem. Let {Pn } ⊂ L(Lp (V )) be a sequence of finite dimensional projections converging strongly to the identity. By noting that Pn ψ → ψ uniformly on compact subsets of Lp (V ), we have Pn K(x)ψ → K(x)ψ in Lp (V ) uniformly in ψ ∈ B and x ∈ Ω, where B is the unit ball of Lp (V ), i.e. sup kK(x) − Pn K(x)kL(Lp (V )) → 0. x∈Ω

In view of (5), the collision operator K is approximated in L(Lp (Ω × V )) as close as we want by finite dimensional (with respect to velocities) collision operators of the form X Lp (Ω × V ) 3 ϕ 7→ hK(x)ϕ(x), e0i iei i∈I

where {ei ; i ∈ I} is a basis of the finite dimensional range of Pn and {e0i ; i ∈ I} ⊂ 0 Lp (V ) is the dual basis ½ 1 if i = j, 0 hei , ej i = 0 otherwise. Notice that the finite dimensional collision operator has the form X XZ ei (v)ei (x, v 0 )dµ(v 0 ), Lp (Ω × V ) 3 ϕ 7→ hϕ(x), K 0 (x)e0i iei = i∈I

i∈I V 0

where ei (x, v 0 ) = K 0 (x)e0i . Let {Pn0 } ⊂ L(Lp (V )) be a sequence of finite dimensional projections converging strongly to the identity. By noticing that Pn0 ψ 0 → ψ 0 uniformly 0 on compact subsets of Lp (V ) and that {K 0 (x)e0i ; x ∈ Ω} is relatively compact, ° ° sup °K 0 (x)e0i − Pn0 K 0 (x)e0i °Lp0 (V ) → 0, x∈Ω

n 0 o P 0 00 0 00 0 hK 0 (x)e0i , fj ifj = hei , K(x)fj ifj , fj ; j ∈ J is a basis of j∈J n 00j∈J o 0 the finite dimensional range of Pn and fj ; j ∈ J ⊂ Lp (V ) is a dual basis. We put where Pn0 K 0 (x)e0i =

P

00

αj = he0i , K(x)fj i. 8

¯ 00 00 ¯ Since {K(x)fj ; x ∈ Ω} is relatively compact in Lp (V ), we have sup¯he0i , K(x)fj i¯ < ∞ x∈Ω

and then αj ∈ L∞ (Ω) . Finally the uniform continuity of αj is a consequence of (6). ¥ Lemma 2 [10]. Let µ be a finite Radon measure on RN such that the affine hyperplanes (i.e. the translated hyperplanes) have zero µ-measure. Then ¯ ¯ © ª sup µ ⊗ µ (v, v 0 ) ∈ RN × RN ; ¯(v − v 0 ).e¯ < ε → 0 as ε → 0. e∈S n−1

We now give the main result of this section. Theorem 8 Let 1 < p < +∞. We assume that the affine hyperplanes have zero µmeasure and that the collision operator is regular. (a) If Ω = RN and if RN 3 x 7→ hK(x)ϕ, ψi 0

is uniformly continuous for every (ϕ, ψ) ∈ Lp (V ) × Lp (V ), then 0 < t 7→ R1 (t) ∈ L(Lp (RN × V )) is continuous in operator norm. (b) If Ω RN and if Ω 3 x 7→ hK(x)ϕ, ψi 0

is uniformly continuous for every (ϕ, ψ) ∈ Lp (V ) × Lp (V ) and its extension (by continuity) to Ω vanishes on ∂Ω, then 0 < t 7→ R1 (t) ∈ L(Lp (Ω × V )) is continuous in operator norm. Before giving the proof, we observe that Theorems 1(c), 2, 7 and 8, and Remark 2 imply the following partial spectral mapping theorem. Corollary 1 Under the assumptions of Theorem 8 we have : (i) If Ω = RN , then σcrit (V (t)) = σcrit (U (t)) and

n o ∗∗ ∗ σ(V (t))∩ µ; |µ| < e−tλ or |µ| > e−tλ = et(σ(T +K)∩{λ; (ii) If Ω

Reλ−λ∗ })

RN , then σcrit (V (t)) = σcrit (U (t))

and

n o ∗ σ(V (t)) ∩ µ; |µ| > e−tλ = et(σ(T +K)∩{λ;

9

Reλ>−λ∗ })

.

.

Proof of Theorem 8 : The proof of Part (a) is quite technical and is given in two steps (Lemma 3 and Lemma 4 below). We will show also how Part (b) can be deduced from Part (a). Proof of Part (a): We recall that R1 (t) is continuous in t > 0 for the uniform topology if and only if Zt U1 (t) = U (t − s)KU (s)ds 0

is [8, Theorem 2.7, p. 18]. Since U1 (t) depends linearly and continuously, in the operator norm topology, on the collision operator K then, according to Lemma 1, there exists a sequence of collision operators Kn of separable form such that Zt U (t − s)Kn U (s)ds → U1 (t)

(n → ∞)

0

in operator norm and the convergence is uniform in t ∈ [0, T ] for all T > 0. Then, without loss of generality, we may restrict ourselves to a collision operator with a kernel of the form Z p K : L (Ω × V ) 3 ϕ 7→ α(x) f (v)g(v 0 )ϕ(x, v 0 )dµ(v 0 ), V

Lp (V

0 Lp (V

L∞ (Ω)

where f ∈ ), g ∈ ) and α ∈ is uniformly continuous. By density again, we may assume that f and g are continuous with compact supports. In this case U1 (t) maps Lq (RN × V ) into itself for all q ≥ 1 and consequently, by interpolation arguments, we can restrict ourselves to the case where p = 2. Let U∞ (t) : L2 (RN × V ) 3 ϕ 7→ e−tσ(v) ϕ(x − tv) ∈ L2 (RN × V ). We have

Zt U1 (t)ϕ =

U∞ (t − s)KU∞ (s)ϕds. 0

A simple calculation shows that Zt U1 (t)ϕ =

U∞ (t − s)KU∞ (s)ϕds 0

Zt

Z −(t−s)σ(v)

=

e 0

V

Zt

Z −(t−s)σ(v)

=

e 0

0

α(x − (t − s)v)ϕ(x − (t − s)v − sv 0 , v 0 )e−sσ(v ) g(v 0 )dµ(v 0 )ds

f (v)

0

ψs (x − (t − s)v − sv 0 , v 0 )e−sσ(v ) g(v 0 )dµ(v 0 )ds,

f (v) V

10

(7)

where ψs is related to ϕ by the relation ψs (x, v) = α(x + sv)χW (v)ϕ(x, v), where s ∈ [0, T ] (T is fixed) and χW is the indicator function of W := supp f We will see in the proof of Lemma 4 that

S

supp g.

s 7→ αs χW ∈ L∞ (Ω × V ) is continuous, where αs χW : (x, v) 7→ α(x + sv)χW (v), so ψ : s 7→ ψs belongs to C([0, T ]; L2 (RN × V )). We note that the term (7) depends continuously on ψ. Write Z 1 cs (ξ, v)eix.ξ dξ, ψs (x, v) = ψ (2π)n/2

(8)

RN

cs (·, ·) is the L2 -Fourier transform of ψs with respect to the space variable and where ψ where the meaning of (8) is Z 1 cs (ξ, v)eix.ξ dξ ψ ψs (·, ·) = lim M →∞ (2π)n/2 |ξ|≤M

in the norm of C([0, T ]; L2 (RN × V )) and kψs (·, ·)k2L2 (Rxn ×V )

Z Z ¯ ¯2 ¯c ¯ = ¯ψs (ξ, v 0 )¯ dξdµ(v 0 ) RN V

for s ∈ [0, T ] . Thus, we can write (7) as the limit (M → ∞) of 1 (2π)n/2

Zt

Z e

−(t−s)σ(v)

0

Z 0

0

cs (ξ, v 0 )g(v 0 )e−sσ(v ) e−i((t−s)v+sv ).ξ dξdµ(v 0 )ds, eix.ξ ψ

f (v) V |ξ|≤M

which is equal to Z

1 (2π)n/2

Zt e

ix.ξ

Z

0

|ξ|≤M

cs (ξ, v 0 )g(v 0 )e−sσ(v0 ) e−i((t−s)v+sv0 ).ξ dµ(v 0 )dsdξ ψ

e−(t−s)σ(v) f (v) V

by Fubini’s theorem since the function under the integral signs is integrable with respect to all variables. Let OM (t) be the operator on L2 (RN × V ) Zt

Z ϕ 7→

e |ξ|≤M

ix.ξ

Z e

0

−(t−s)σ(v)

cs (ξ, v 0 )g(v 0 )e−sσ(v0 ) e−i((t−s)v+sv0 ).ξ dµ(v 0 )dsdξ. ψ

f (v) V

The following two lemmas will end the proof of Part (a). 11

Lemma 3 OM (t) converges to U1 (t), as M goes to ∞, in operator norm and uniformly in t ∈ [0, T ]. Lemma 4 The map 0 < t 7→ OM (t) ∈ L(L2 (RN × V )) is continuous in operator norm. Proof of Lemma 3 : It is equivalent to show that Z

Zt e

ix.ξ

|ξ|>M

Z e

−(t−s)σ(v)

cs (ξ, v 0 )g(v 0 )e−sσ(v0 ) e−i((t−s)v+sv0 ).ξ dµ(v 0 )dsdξ → 0 ψ

f (v)

0

V

in L2 (RN × V ) as M → ∞ uniformly in ϕ bounded in L2 (RN × V ) and t ∈ [0, T ]. Let G : [0, T ] 3 s 7→ G(s) ∈ L∞ (RN × V ), where G(s) : RN × V 3 (x, v) 7→ α(x + sv)χW (v) ∈ R. Note that G ∈ C([0, T ]; L∞ (RN × V )) because kG(s) − G(¯ s)k∞ = ess sup |χW (v)| |α(x + sv) − α(x + sv)| , α is uniformly continuous, W is bounded and |(x + sv) − (x + sv)| ≤ C |s − s| . Note that ψs = G(s)ϕ ∈ C([0, T ]; L2 (RN × V )). Note that G can be approximated uniformly by a sequence of step functions : There exists {kn } ⊂ N such that for all i = 1, ..., kn , there exists an interval Iin included in [0, T ] and ψin ∈ L∞ (RN × V ) such that kn ° ° X ° ° sup °G(s) − χIin (s)ψin °

s∈[0,T ]

→ 0.

L∞ (RN ×V ) n→∞

i=1

It follows that kn ° ° X ° ° sup °ψs − χIin (s)ψin ϕ°

s∈[0,T ]

L2 (RN ×V )

i=1

kn ° ° X ° ° = sup °G(s)ϕ − χIin (s)ψin ϕ° s∈[0,T ]

→ 0,

L2 (RN ×V ) n→∞

i=1

uniformly in kϕkL2 (RN ×V ) ≤ 1. By Parseval’s Identity we have kn ° ° °c X n n ϕ° d (s) ψ χ − sup °ψ s Ii i °

s∈[0,T ]

L2 (RN ×V )

i=1

kn ° ° X ° ° χIin (s)ψin ϕ° = sup °ψs − s∈[0,T ]

i=1

→ 0,

L2 (RN ×V ) n→∞

the convergence being uniform in kϕkL2 (RN ×V ) ≤ 1. Now Z

Zt dξe

|ξ|>M

ix.ξ

Z cs (ξ, v 0 )g(v 0 )e−sσ(v0 ) e−i((t−s)v+sv0 ).ξ dµ(v 0 )ds ψ

e−(t−s)σ(v)

f (v) 0

V

12

is equal to Z

Zt dξe

ix.ξ

f (v)

Z e

0

|ξ|>M

cs (ξ, v 0 ) − (ψ

−(t−s)σ(v)

kn X

0 0 −sσ(v 0 ) n χIin (s)ψd i ϕ(ξ, v ))g(v )e

i=1

V 0

×e−i((t−s)v+sv ).ξ dµ(v 0 )ds Zt Z ix.ξ −(t−s)σ(v) 0 0 −sσ(v 0 ) n dξe f (v) e χIin (s)ψd i ϕ(ξ, v )g(v )e

Z kn X

+

i=1

0

|ξ|>M

×e =:

0,n OM (t)ϕ

+

kn X

V

−i((t−s)v+sv 0 ).ξ

dµ(v 0 )ds

i,n OM (t)ϕ.

i=1

° 0,n ° Let us show that °OM (t)° → 0 as n → ∞ uniformly in t ∈ [0, T ] and uniformly in M . °2 ° 0,n By Parseval’s Identity °OM (t)ϕ° is equal to Z =

Z

Zt Z kn ¯ X ¯ −(t−s)σ(v) c 0 0 0 n dξ ¯f (v) e (ψs (ξ, v ) − χIin (s)ψd i ϕ(ξ, v ))g(v )

dµ(v) V

0 V

|ξ|>M

¯2 0 0 ¯ ×e−sσ(v ) e−i((t−s)v+sv ).ξ dµ(v 0 )ds¯ .

By the Cauchy-Schwarz Inequality, we  t Z Z Z Z  dµ(v) dξ V

|ξ|>M

Zt ×

Zt dµ(v) Z

×

Z ¯ ¯2 ¯ −(t−s)σ(v) −sσ(v0 ) 0 ¯ e g(v )f (v)¯ dµ(v 0 )ds ¯e

Zt Z ¯ kn ¯2 X ¯ c 0 n ϕ(ξ, v 0 )¯ dξdsdµ(v 0 ) χIin (s)ψd ¯(ψs (ξ, v ) − ¯ i

|ξ|>M 0 V

V

0 V

0 V

V

≤T

¯ ¯2 ¯ −(t−s)σ(v) −sσ(v0 ) 0 ¯ e g(v )f (v)¯ dsdµ(v 0 ) ¯e

i=1

Z

Z

° 0,n °2 majorize °OM (t)ϕ° by

 Z ¯ kn ¯2 X ¯ c 0 ¯ 0  n χIin (s)ψd ¯(ψs (ξ, v 0 ) − i ϕ(ξ, v ))¯ dsdµ(v )

0 V

≤

i=1

i=1

Zt Z ¯ kn ° ° ¯2 °c X n d ° ¯ −(t−s)σ(v) −sσ(v0 ) 0 ¯ 0 χIi ψin ϕ° 2 . e g(v )f (v)¯ dsdµ(v ) sup °ψs − dµ(v) ¯e s∈[0,T ]

0 V

i=1

L

The last term tends to zero as n tends to infinity, uniformly in kϕkL2 (RN ×V ) ≤ 1, uniformly in t ∈ [0, T ] and also uniformly in M. Thus, it suffices to choose n large enough 13

kn kn °P ° P i,n i,n and to show that ° OM (t)° → 0 as M → ∞ uniformly in t ∈ [0, T ] . Since OM (t) i=1 i=1 ° i,n °2 consists of n similar terms, it suffices to deal with one of them. Thus °OM (t)ϕ° is equal to

Z

Z dµ(v)

V

¯ Z ¯ dx¯

RN

Zt ix.ξ

dξe

Z 0

0 0 −sσ(v ) n χIin (s)ψd i ϕ(ξ, v )g(v )e

e−(t−s)σ(v)

f (v) 0

|ξ|>M

V

¯2 ¯ ×e dµ(v 0 )ds¯ Z Z Z ¯ Zt ¯ 0 0 −sσ(v 0 ) −(t−s)σ(v) n = dµ(v) dξ ¯ e f (v) χIin (s)ψd i ϕ(ξ, v )g(v )e −i((t−s)v+sv 0 ).ξ

V

Z =

Z

V

Z

V

Z

−(t−s)σ(v)

e

[0,t]∩Iin

×e

Z 0 0 −sσ(v 0 ) n ψd i ϕ(ξ, v )g(v )e

f (v) V

−i((t−s)v+sv 0 ).ξ

dξ

¯2 ¯ dµ(v 0 )ds¯

Z ¯ ¯ −tσ(v) ¯e

¯2 ¯ dµ(v 0 )ds¯ Z ¯2 0 0 ¯ e−s(σ(v )−σ(v)) f (v)g(v 0 )e−is(v −v).ξ ds¯ dµ(v 0 ) n

[0,t]∩Ii Z V¯ ¯2 ¯d 0 ¯ n × ¯ψi ϕ(ξ, v )¯ dµ(v 0 )

|ξ|>M

Z sup |ξ|>M

≤

¯ ¯ dξ ¯

dµ(v)

V

−i((t−s)v+sv 0 ).ξ

|ξ|>M

Z

≤

×e

dµ(v)

≤

0

|ξ|>M

V

Z ¯ ¯ dµ(v) ¯e−tσ(v)

V

Z e

V

[0,t]∩Iin

V

[0,t]∩I n

−s(σ(v 0 )−σ(v))

0

f (v)g(v )e

−is(v 0 −v).ξ

¯2 ¯ ds¯ dµ(v 0 )

° ° n ° ×°ψd i ϕ L2 (RN ×V ) Z Z ¯ Z ¯2 0 0 ¯ −tσ(v) ¯ sup dµ(v) ¯e e−s(σ(v )−σ(v)) f (v)g(v 0 )e−is(v −v).ξ ds¯ dµ(v 0 ) |ξ|>M

V

° °2 ° °2 i ×°ψin °∞ °ϕ°L2 (RN ×V ) .

We have to show that Z Z ¯ ¯ lim dµ(v)dµ(v 0 )¯e−tσ(v) f (v)g(v 0 ) |ξ|→∞

Z

¯2 0 0 e−s(σ(v )−σ(v)) e−is(v −v).ξ ds¯ = 0,

[0,t]∩Iin

V V

uniformly in t ∈ [0, T ]. Since f and g are continuous with compact supports, this amounts to Z Z ¯2 ¯ Z 0 0 ¯ 0 ¯ e−s(σ(v )−σ(v)) e−is(v −v).ξ ds¯ = 0 lim dµ(v)dµ(v )¯ |ξ|→∞

W W

[0,t]∩Iin

14

S uniformly in t ∈ [0, T ] where W = supp f supp g. To prove this, introduce polar N −1 coordinates ξ = |ξ| e with e ∈ S . Keeping in mind that (v, v 0 ) ∈ W × W, we note that for every ε > 0 Z Z ¯ Z ¯2 0 0 ¯ 0 ¯ dµ(v)dµ(v )¯ e−s(σ(v )−σ(v)) e−is(v −v).ξ ds¯ W W

ZZ

=

Iin ∩[0,t]

Z

¯ ¯ ¯

¯2 0 0 ¯ e−s(σ(v )−σ(v)) e−is|ξ|e.(v −v) ds¯ dµ(v)dµ(v 0 )

|e.(v 0 −v)|≤ε Iin ∩[0,t]

ZZ

Z

¯ ¯ ¯

+

¯2 0 0 ¯ e−s(σ(v )−σ(v)) e−is|ξ|e.(v −v) ds¯ dµ(v)dµ(v 0 )

|e.(v 0 −v)|>ε Iin ∩[0,t]

≤ T

ZZ

−2s(σ(v)−σ(v 0 ))

ess sup

e

s∈[0,T ];v,v 0 ∈W

ZZ

¯ ¯ ¯

+ |e.(v 0 −v)|>ε

dµ(v)dµ(v 0 )

sup e∈S n−1 |e.(v 0 −v)|≤ε

Z e

−s(σ(v 0 )−σ(v)) −is|ξ|e.(v 0 −v)

e

¯2 ¯ ds¯ dµ(v)dµ(v 0 ).

Iin ∩[0,t]

According to Lemma 2,

ZZ dµ(v)dµ(v 0 )

sup e∈S n−1 |e.(v 0 −v)|≤ε

is arbitrarily small for ε small enough. We fix ε small enough and consider the second term. Consider the set © ª S = St,v,v0 ∈ L1 (R), (t, v, v 0 ) ∈ [0, T ] × W × W , where

0

St,v,v0 : R 3 s 7→ χIin ∩[0,t] (s)e−s(σ(v )−σ(v)) ∈ R. The set S is relatively compact in L1 (R). Indeed, let (tp , vp , vp0 )p∈N be a sequence in [0, T ]×W ×W. We can extract a subsequence (tpk , vpk , vp0 k )k∈N such that (tpk , σ(vpk ), σ(vp0 k ))k∈N converges to (t∗ , a, b). By the dominated convergence theorem the sequence (Stpk ,vpk ,vp0 )k∈N k converges in L1 (R) to R 3 s 7→ χIin ∩[0,t∗ ] (s)e−s(b−a) . By Riemann-Lebesgue’s theorem and the compactness of S we have Z St,v,v0 (s)e−isτ ds = 0 lim |τ |→∞

R

uniformly in (t, v, v 0 ) ∈ [0, T ] × W × W. In particular, since |e.(v 0 − v)| > ε, ¯ ¯ Z 0 0 ¯ ¯ e−s(σ(v )−σ(v)) e−is|ξ|e.(v −v) ds¯ = 0 lim ¯ |ξ|→∞

Iin ∩[0,t]

15

uniformly in (t, v, v 0 ) ∈ [0, T ] × W × W and consequently ZZ ¯ Z ¯2 0 0 ¯ ¯ lim e−s(σ(v )−σ(v)) e−is|ξ|e.(v −v) ds¯ dµ(v)dµ(v 0 ) = 0 ¯ |ξ|→∞ |e.(v 0 −v)|>ε Iin ∩[0,t]

uniformly in t ∈ [0, T ].

¥

Proof of Lemma 4: Let t > 0. Then OM (t)ϕ − OM (t)ϕ is equal to Z

Zt ix.ξ

dξe

f (v)

Z e

cs (ξ, v 0 )g(v 0 )e−sσ(v0 ) e−i((t−s)v+sv0 ).ξ dµ(v 0 )ds ψ

−(t−s)σ(v)

0

|ξ|≤M

Z

V

Zt ix.ξ

−

dξe

f (v)

Z e

0

|ξ|≤M

Z

cs (ξ, v 0 )g(v 0 )e−sσ(v0 ) e−i((t−s)v+sv0 ).ξ dµ(v 0 )ds ψ

−(t−s)σ(v) V

Zt

Z ix.ξ

=

0

dξe 

 +

f (v)g(v )e V

|ξ|≤M

0

0

cs (ξ, v 0 )dµ(v 0 )ds e−s(σ(v )−σ(v)) e−is(v −v).ξ ψ

−tσ(v)−itv.ξ t

Z

Z dξeix.ξ

f (v)g(v 0 )(e−tσ(v)−itv.ξ − e−tσ(v)−itv.ξ ) V

|ξ|≤M



Zt

0 0 cs (ξ, v 0 )dµ(v 0 )ds e−s(σ(v )−σ(v)) e−is(v −v).ξ ψ

× 0 1 2 =: OM (t, t)ϕ + OM (t, t)ϕ.

Let us show that

° 1 ° ° 2 ° °OM (t, t)° → 0 and °OM (t, t)° → 0. t→t

t→t

° 1 °2 We note that °OM (t, t)ϕ° is equal to Z

Z dµ(v)

V

¯ Z ¯ dx¯

RN

Z ix.ξ

dξe

|ξ|≤M

Zt 0

f (v)g(v )e

−tσ(v)−itv.ξ

V

t ¯2 ¯ 0 0 c ×e ψs (ξ, v )dµ(v )ds¯ Z Z Zt ¯Z 0 ¯ 0 −tσ(v)−itv.ξ dξ ¯ f (v)g(v )e = dµ(v) e−s(σ(v )−σ(v)) −is(v 0 −v).ξ

V

|ξ|≤M

V

t

¯2 0 cs (ξ, v 0 )dµ(v 0 )ds¯¯ ×e−is(v −v).ξ ψ 16

0

e−s(σ(v )−σ(v))

Z

Z ≤

Z Zt ¯ ¯2 ¯ 0 −tσ(v) −s(σ(v 0 )−σ(v)) ¯ dξ e ¯ dsdµ(v 0 ) ¯f (v)g(v )e

dµ(v) V

V

|ξ|≤M

t

Z Zt ¯ ¯2 ¯c 0 ¯ × ¯ψs (ξ, v )¯ dsdµ(v 0 ) V

ZZ

t

° °2 C1 dµ(v)dµ(v 0 ) kαk2∞ °ϕ°L2 (RN ×V ) → 0 as t → t¯

≤ (t − t)2 W ×W

uniformly in kϕkL2 (RN ×V ) ≤ 1 where C1 =

¯ ¯2 0 ¯ ¯ ess sup ¯e−tσ(v) e−s(σ(v)−σ(v )) f (v)g(v 0 )¯ . W ×W,s∈[0,T ]

Now, ° 2 ° °OM (t, t)ϕ°2 Z Z ¯ Z ¯ = dµ(v) dx¯ V

RN

×

=

V

|ξ|≤M

×e

≤

Z

−is(v 0 −v).ξ

|ξ|≤M

×e ZZ

0

¯2 cs (ξ, v 0 )dsdµ(v 0 )¯¯ ψ

Z Zt ¯ ¯ dξ ¯f (v)g(v 0 )(e−tσ(v)−itv.ξ − e−tσ(v)−itv.ξ )

dµ(v) V

¯2 0 0 cs (ξ, v 0 )dsdµ(v 0 )¯¯ e−s(σ(v )−σ(v)) e−is(v −v).ξ ψ

Zt ¯Z 0 ¯ 0 −tσ(v)−itv.ξ −tσ(v)−itv.ξ dξ ¯ f (v)g(v )(e −e ) e−s(σ(v )−σ(v))

dµ(v)

Z

V

0

Z

V

f (v)g(v 0 )(e−tσ(v)−itv.ξ − e−tσ(v)−itv.ξ )

dξe

|ξ|≤M

Zt

Z

Z ix.ξ

V

0

Z Zt ¯ ¯2 ¯c ¯ ¯ dsdµ(v ) ¯ψs (ξ, v 0 )¯ dsdµ(v 0 )

¯2 −s(σ(v 0 )−σ(v)) ¯

0

0

V

¯2 ZT ¯ 0 ¯ −tσ(v)−itv.ξ −tσ(v)−itv.ξ ¯ dµ(v)dµ(v )¯e −e e−2s(σ(v )−σ(v)) ds ¯ 0

≤ C2 sup

ξ≤M W ×W

0

° °2 ° ° ×T °α°∞ °ϕ°L2 (RN ×V ) ,

where C2 = ess sup | f (v)g(v 0 ) |2 . One sees that for all A > 0 W ×W

ZZ

¯ ¯2 ZT 0 ¯ −tσ(v)−itv.ξ −tσ(v)−itv.ξ ¯ e−2s(σ(v )−σ(v)) ds → 0 as t → t dµ(v)dµ(v )¯e −e ¯ 0

0

W ×W

17

° 2 ° uniformly in | ξ |≤ A and consequently °OM (t, t)° → 0. t→t

¥

Proof of Theorem 8 Part (b) : As for Part (a) it suffices to prove that 0 < t 7→ U1 (t) is continuous for the uniform topology. By the convexity of Ω we have U (t)ϕ = RU∞ (s)Eϕ, where E is the trivial extension by zero to L2 (RN × V ) and R : L2 (RN × V ) → L2 (Ω × V ) is the restriction operator. Then Zt U1 (t)ϕ =

RU∞ (t − s)EKRU∞ (s)Eϕds, 0

and it suffices to show the continuity in operator norm of Zt 0 ≤ t 7→

U∞ (t − s)EKRU∞ (s)ds. 0

Let K∞ := EKR. We note that for ϕ ∈ L2 (RN × V ) ½ K(x)ϕ(x) on Ω, K∞ (x)ϕ(x) = 0 on RN \Ω. It follows that for ϕ ∈ L2 (V ) and ψ ∈ L2 (V ) ½ hK(x)ϕ, ψi hK∞ (x)ϕ, ψi = 0

if x ∈ Ω, if x ∈ /Ω

is uniformly continuous in x ∈ RN since hK(x)ϕ, ψi vanishes at the boundary ∂Ω. Dealing now with K∞ instead of K, the norm continuity of 0 ≤ t 7→ U1 (t) ∈ L(Lp (Ω × V )) is a consequence of Part (a).

¥

We show now that in Theorem 8 the assumption on µ is optimal. Theorem 9 Let µ be a Radon measure with compact support V and let Ω = RN . We assume that σ = 0 and that Z 2 N K : L (R × V ) 3 ϕ 7→ ϕ(x, v 0 )dµ(v 0 ). V

© ª If there exists a hyperplane H = v ∈ RN , v.e = c with positive µ−measure where e ∈ S n−1 and c ∈ R∗ , then there exists a sequence tn → 0 such that t 7→ R1 (t) is not continuous in norm topology at tn for all n. 18

Proof : If (for some T > 0) ]0, T ] 3 t 7→ R1 (t) is continuous in operator norm, then ]0, T ] 3 t 7→ U1 (t) would be also continuous [8, Lemma 2.3, p. 16]. Thus it suffices to prove that t 7→ U1 (t) is not continuous in norm topology for t > 0. Let t > 0, put δ = sup |v| and v∈V

r = δ(t¯ + 1). Denoting by Ω0 (r) (respectively Ω0 (3r)) the ball centred in 0 with radius r (respectively 3r). We have x − (t − s)v − sv 0 ∈ Ω0 (3r) for x ∈ Ω0 (r), 0 ≤ s ≤ t and t ≤ t + 1. Let ϕm ∈ L2 (RN × V ) ϕm = fm (x)h(v) where ½ imx.e e if x ∈ Ω0 (3r) fm (x) = 0 otherwise, R and h ∈ L2 (H) such that h(v 0 )dµ(v 0 ) > 0. H

Recalling that

Zt Z fm (x − (t − s)v − sv 0 )h(v 0 )dµ(v 0 )ds,

U1 (t)ϕm = 0 V

then we have for t ≤ t + 1 ° ° °U1 (t)ϕm − U1 (t)ϕm °2 2 N L (R ×V ) Z =

Z

¯ Zt Z ¯ dx¯ fm (x − (t − s)v − sv 0 )h(v 0 )dµ(v 0 )ds

dµ(v) V

0 V

RN

Zt Z −

¯2 ¯ fm (x − (t − s)v − sv 0 )h(v 0 )dµ(v 0 )ds¯

0 V

≥

1 2

Z

Z

¯ ¯2 ¯ Z Zt ¯2 0 ¯ −imtv.e ¯ −imtv.e ¯ ¯ dx¯e −e e−ims(v−v ).e h(v 0 )dsdµ(v 0 )¯ ¯ ¯

dµ(v) H

H 0

Ω0 (r)

Z

¯ ¯2 ¯ Z Zt ¯2 ¯ −imtc ¯ −imtc ¯ ¯ dµ(v)¯e −e h(v 0 )dsdµ(v 0 )¯ ¯ ¯

H

Z

≥

1 |Ω0 (r)| 2

≥

1 |Ω0 (r)| t2 2

H

H 0

¯Z ¯2 ¯ ¯2 ¯ ¯ ¯ ¯ dµ(v)¯ h(v 0 )dµ(v 0 )¯ ¯e−imtc − e−imtc ¯ . H

Choosing the sequence tm = t +

π , we obtain cm

° ° π °U1 (tm )ϕm − U1 (t)ϕm °2 2 N ≥ 2 |Ω0 (r)| min{t¯2 , (t¯ + )2 } L (R ×V ) c

Z H

19

¯Z ¯2 ¯ ¯ dµ(v)¯ h(v 0 )dµ(v 0 )¯ H

and consequently there exists a constant C > 0 such that ° ° °U1 (tm ) − U1 (t)° ≥ C for all m ∈ N∗ .

5

¥

A spectral mapping theorem

In Theorem 8, the assumption on the velocity measure µ is optimal. On the other hand, the dependence of the scattering kernel on the spatial variable is restrictive. Indeed, assumption (6) means that Ω 3 x 7→ K(x) ∈ L(Lp (V )) is ”smooth” and vanishes at the boundary ∂Ω. Such an assumption excludes for instance the piecewise constant (with respect to x ∈ Ω) scattering kernels used in nuclear reactor theory or even the space homogeneous scattering kernels when Ω $ RN . In the present section, we introduce a stronger assumption on the velocity measure µ : Z eiz.v dµ(v) → 0 as |z| → ∞ (9) D

for all Borel set D ⊂ RN with µ(D) < ∞. This assumption allows us first to avoid the restrictive condition (6) and secondly to obtain full spectral mapping theorems. We note that (9) is still very general and covers in particular the usual continuous or multigroup models. We start with the following observation. Lemma 5 Let µ be a positive Radon measure satisfying (9). Then the affine hyperplanes have zero µ-measure. Proof : Suppose there exists some affine hyperplane H such that µ(H) > 0. Let v ∈ H. There exists e ∈ S N −1 such that H = {v ∈ RN ; (v − v).e = 0}. Let D ⊂ H be such that 0 < µ(D) < ∞. Then, for all z = te (t ∈ R), Z Z iz.v e dµ(v) = eite.v dµ(v) D Z

D

eite.(v−v) eite.v dµ(v)

= D

= eite.v µ(D),

20

so

¯Z ¯ ¯ ¯ iz.v e dµ(v) ¯ ¯ = µ(D) > 0 for all z ∈ Re. D

¥ We now give a basic result. Lemma 6 Let 1 < p < ∞ and let Ω be convex. We assume that the collision operator K is regular and that (9) is satisfied. Then 0 < t 7→ R2 (t) ∈ L(Lp (Ω × V )) is continuous in operator norm. Proof : We consider first the case p = 2. We recall that 0 < t 7→ R2 (t) is continuous for the uniform topology if and only if 0 < t 7→ [U K]2 (t) is continuous for the uniform topology (see [8, Theorem 2.7, p. 18]). On the other hand [U K]2 = (U K) ∗ (U K) = U ∗ (KU K), where ∗ is the time convolution which associates to strongly continuous (operator valued) mappings f, g : [0, +∞[ → L(L2 (Ω × V )) the strongly continuous mapping Zt f (t − s)g(s)ds ∈ L(L2 (Ω × V ))

f ∗ g : [0, +∞[ 3 t 7→ 0

(see [8, p. 14]). On the other hand, 0 < t 7→ [U K]2 (t) is continuous in the uniform topology if 0 < t 7→ KU (t)K is [8, Lemma 2.3, p. 16]. Moreover, since the collision operator is regular, we may argue by approximation and restrict ourselves to K1 U (t)K2 where Ki (i = 1, 2) have kernels of the form ki (x, v, v 0 ) = αi (x)fi (v)gi (v 0 ); αi ∈ L∞ (Ω); fi , gi ∈ L2 (Ω). We note that K1 U (t)K2 is factorizable as e K1 U (t)K2 = O1 S(t)O 2, where

Z O2 : L2 (Ω × V ) 3 ϕ 7→ α2 (x) Z e : L2 (Ω) 3 ψ 7→ S(t)

ϕ(x, v 0 )g2 (v 0 )dµ(v 0 ) ∈ L2 (Ω), V

h(v)e−tσ(v) ψ(x − tv)χ(t ≤ τ (x, v))dµ(v) ∈ L2 (Ω) V

21

(where h(v) = g1 (v)f2 (v)) and O1 : L2 (Ω) 3 ψ 7→ α1 (x)f1 (v)ψ(x) ∈ L2 (Ω × V ). e (t > 0) depends continuously on t > 0 in the uniform Hence it suffices to prove that S(t) topology. By approximating h ∈ L1 (V ; dµ) we may suppose that h is continuous with compact support. Since Ω is convex, e = RSe∞ (t)E S(t) where E is the trivial extension, R is the restriction and Z 2 N e S∞ (t) : L (R ) 3 ψ 7→ h(v)e−tσ(v) ψ(x − tv)dµ(v) ∈ L2 (RN ). V

Let us show that Se∞ (t) is continuous in t > 0 for the uniform topology. For ψ ∈ L2 (RN ), Z e S∞ (t)ψ = h(v)e−tσ(v) ψ(x − tv)dµ(v) Z = ψ(x − z)dνt (z) = ψ ∗ νt , where νt is the image of Radon measure h(v)e−tσ(v) dµ under the dilation v 7→ tv. Let t > 0 be fixed. Then ° ° ° ° °Se∞ (t)ψ − Se∞ (t)ψ ° 2 N = °ψ ∗ (νt − ν )° 2 N t L (R ) L (R ) sup |νbt (ζ) − νbt (ζ)| kψkL2 (RN ) ,

≤

ζ∈RN

whence

° ° °Se∞ (t) − Se∞ (t)° 2 N ≤ sup |νbt (ζ) − νbt (ζ)| . L(L (R )) ζ∈RN

On the other hand

Z

Z e−itv.ζ h(v)e−tσ(v) dµ(v) −

νbt (ζ) − νbt (ζ) = RN

e−itv.ζ h(v)e−tσ(v) dµ(v)

RN

splits as Z

Z e

−itv.ζ

h(v)e

RN

−tσ(v)

RN

Z −

e−itv.ζ h(v)(e−tσ(v) − e−tσ(v) )dµ(v)

dµ(v) +

e−itv.ζ h(v)e−tσ(v) dµ(v)

RN

=: I1 (ζ) + I2 (ζ) − I3 (ζ). 22

It is clear that I2 (ζ) → 0 as t → t uniformly in ζ ∈ RN . Moreover, I1 (ζ) and I3 (ζ) are arbitrarily small for |ζ| large enough, uniformly in t belonging to a fixed small neighborhood of t, because the Fourier Transform of h(v)e−t¯σ(v) dµ(v) goes to zero at infinity. Moreover, for all A > 0 it is clear that I1 (ζ) − I3 (ζ) → 0 as t → t uniformly in | ζ |≤ A. Hence ° ° °e ° °S∞ (t) − Se∞ (t)° 2 N → 0 as t → t, L(L (R ))

which ends the proof for p = 2. The case 1 < p < ∞ is tackled as follows. We first note that R2 (t) ∈ L(Lp (Ω×V )) depends continuously on K ∈ L(Lp (Ω × V )) (uniformly in t ∈ [0, T ] for all T > 0). Therefore it suffices to give the proof for a smooth collision operator K, say with a kernel of the form X k(x, v, v 0 ) = αi (x)fi (v)gi (v 0 ) i∈I

L∞ (Ω)

where I is finite, αi ∈ and fi , gi are continuous with compact supports. In such a case R2 (t) ∈ L(Lr (Ω × V )) for all 1 ≤ r ≤ ∞ and consequently the above L2 results extends to Lp spaces (1 < p < ∞) by interpolation. ¥ Before giving the spectral mapping theorem we need one more preliminary lemma. Lemma 7 Let 1 < p < ∞ and let Ω be convex. We assume that (9) is satisfied and that the collision operator is regular. Then σap (T ) ⊂ σap (T + K). Proof : Let λ ∈ σap (T ). There exists ϕn ∈ D(T ), kϕn k = 1, kλϕn − T ϕn k → 0 as n → +∞. Set ψn,m (x, v) = ϕn (x, v)eim.v where m ∈ ZN is multiindex. We note that kλψn,m − T ψn,m k = kλϕn − T ϕn k shows that kλψn,m − T ψn,m k → 0 as n → +∞ uniformly in m ∈ ZN . Let us show that for each n kKψn,m k → 0 as |m| → ∞. Since K is regular, there exists a sequence (Kl )l∈N of collision operators with kernels of the form X k(x, v, v 0 ) = αil (x)fil (v)gil (v 0 ), i∈I l

23

0

with fil ∈ Lp (V ), gil ∈ Lp (V ) and αil ∈ L∞ (Ω) (I l finite) such that kK − Kl k → 0 as l → ∞. Since kKψn,m k ≤ kKψn,m − Kl ψn,m k + kKl ψn,m k ≤ kK − Kl k + kKl ψn,m k , it suffices to choose l large enough and to show that kKl ψn,m k → 0 as |m| → ∞. Note that Kl is a finite sum of collision operators with kernel of form α(x)f (v)g(v 0 ) and it suffices to show that Z Z ¯Z ¯p ¯ ¯ ¯ α(x)f (v)g(v 0 )ψn,m (x, v 0 )dµ(v 0 )¯ dxdµ(v) → 0 as |m| → ∞, Ω V

V 0

where f ∈ Lp (V ), g ∈ Lp (V ) and α ∈ L∞ (Ω). Now Z Z ¯Z ¯p ¯ ¯ ¯ α(x)f (v)g(v 0 )ψn,m (x, v 0 )dµ(v 0 )¯ dxdµ(v) Ω V

Z Z ¯ VZ ¯p 0 ¯ ¯ = ¯ α(x)f (v)g(v 0 )ϕn (x, v 0 )eim.v dµ(v 0 )¯ dxdµ(v). Ω V

V

Assumption (9) implies the Riemann-Lebesgue property : Z 0 ϕ(v 0 )eiz.v dµ(v 0 ) → 0 as |z| → ∞ ∀ϕ ∈ L1 (RN , dµ). Indeed, this property which is clearly true for simple functions ϕ remains true for ϕ ∈ L1 by density arguments. We note that for almost all (x, v) ∈ Ω × V Z | α(x)f (v)g(v 0 )ϕn (x, v 0 ) | dµ(v 0 ) < +∞, V

because Z Z Ω V

 p Z  | α(x)f (v)g(v 0 )ϕn (x, v 0 ) | dµ(v 0 ) dxdµ(v) < +∞. V

Hence, for almost all (x, v) ∈ Ω × V and for each n, Z 0 α(x)f (v)g(v 0 )ϕn (x, v 0 )eim.v dµ(v 0 ) → 0 as |m| → +∞ V

by the Lebesgue-Riemann property. Moreover ¯Z ¯p 0 ¯ ¯ ¯ α(x)f (v)g(v 0 )ϕn (x, v 0 )eim.v dµ(v 0 )¯ V

Z p

≤ |α(x)f (v)|

V

¯ 0 ¯p0 ¯g(v )¯

Z

¯ ¯ ¯ϕn (x, v 0 )¯p dµ(v 0 ) ∈ L1 (Ω × V )

V

24

so, by the dominated convergence theorem, Z Z ¯Z ¯p 0 ¯ ¯ ¯ α(x)f (v)g(v 0 )ϕn (x, v 0 )eim.v dµ(v 0 )¯ dxdµ(v) → 0 as |m| → +∞. Ω V

V

In particular, for each n there exists mn ∈ ZN such that Z Z ¯Z ¯p 1 0 ¯ ¯ ¯ α(x)f (v)g(v 0 )ϕn (x, v 0 )eimn .v dµ(v 0 )¯ dxdµ(v) ≤ p , n Ω V

so kKψn,mn k ≤

V

1 and consequently n

kλψn,mn − (T + K)ψn,mn k ≤ kλψn,mn − T ψn,mn k + kKψn,mn k 1 ≤ kλψn,mn − T ψn,mn k + → 0 as n → ∞ n so λ ∈ σap (T + K).

¥

We are now ready to give the main result of this section. Theorem 10 Let 1 < p < +∞ and Ω be a convex domain in RN . We assume that the collision operator is regular and that (9) is satisfied. (a) If Ω & RN , then σ(V (t)) = etσ(T +K) ∪ {0} . (b) If Ω = RN , then ∗∗

σ(V (t))∩{µ; | µ |< e−tλ

∗

or | µ |> e−tλ } = et(σ(T +K)∩{λ;

Reλ−λ∗ })

(c) If Ω = RN and if the essential range of σ is connected, then σ(V (t)) = etσ(T +K) . Proof : From Lemma 6 and Theorem 3 we have ωcrit (V (·)) = ωcrit (U (·)). Using Theorems 6 and 7 we get ωcrit (V (·)) = −λ∗ and, by Theorem 1(c), ∗

σ(V (t)) ∩ {µ; | µ |> e−tλ } = et(σ(T +K)∩{λ;

Reλ>−λ∗ })

.

(10)

On the other hand, by Theorem 6, Lemma 7 and the inclusion etσap (T +K) ⊂ σap (V (t)) we obtain ∗

{µ; | µ |≤ e−tλ } ⊂ etσ(T +K) ⊂ σ(V (t))

25

.

which proves the first item. As above, Equality (10) holds when Ω = RN . In this case (U (t))t≥0 can be extended to a positive group with U (−t) : ϕ 7→ etσ(v) ϕ(x + tv) (t ≥ 0). Note that (U (−t))t≥0 is generated by −T and its growth bound is λ∗∗ = ess sup σ. Similarly (V (t))t≥0 can be extended to a group with (V (−t))t≥0 generated by −T − K and can be represented by Dyson-Phillips series. We can show by the same arguments as in Lemma 6 that the corresponding second remainder term depends continuously on t in operator norm. Thus, ωcrit (V (−·)) = ωcrit (U (−·)) = λ∗∗ and

n o tλ∗∗ σ(V (−t)) ∩ µ; |µ| > e = et(σ(−T −K)∩{λ;

Reλ>λ∗∗ })

.

On the other hand µ ∈ σ(V (−t)) if and only if µ−1 ∈ σ(V (t)), so n o ∗∗ ∗∗ σ(V (t)) ∩ µ; |µ| < e−tλ = et(σ(T +K)∩{λ; Reλ 0 for the operator norm if and only if the affine hyperplanes have zero µ-measure. We conjecture that the (partial) spectral mapping theorems fail if some affine hyperplane has a positive µ-measure.

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