Sobolev Regularity in Neutron Transport Theory

The main purpose of this paper is to extend the W 1 ,p regularity results in neutron transport theory, with respect to the Lebesgue measure due to Mokhtar-Kharroubi, (cid:2) 1991 (cid:3) , and to abstract measures covering, in particular, the continuous models or multigroup models. The results are obtained for vacuum boundary conditions as well as periodic boundary conditions. H 2 regularity results are derived when the velocity space is endowed with an appropriate class of measures (cid:2) signed in the multidimensional case (cid:3) .


Introduction
Let D be an open bounded convex subset of R N . Let dμ be a positive bounded measure on R N supported by V {v ∈ R N ; |v| < R}, 0 < R < ∞. We denote and dx is the Lebesgue measure on D.
The streaming operator T is defined Here, n x is the unit outward normal at x ∈ ∂D. It is well known that under the hypothesis dμ {0} 0, any function ψ ∈ X p possesses traces on Γ − see 1-4 . Let k i ·, · i 1, 2 be two measurable functions on V × V such that

1.4
Note that, for all λ > −λ * −dμ ess-inf σ · , the operators N λ K 1 λ − T −1 K 2 and M λ K 1 λ − T −1 map continuously L p dx ⊗ dμ v into itself. In 5 , several W 1,p regularity results of the operators N λ and M λ were established when the velocity space V is endowed with the Lebesque measure. These results play a cornerstone role in the proof of the neutron transport approximations 6-8 . The main purpose of this paper is to extend the results of 5 in two directions. First, we deal with a general class of abstract measures covering, in particular, the continuous models or multigroup models. Secondly, we extend the results to periodic boundary conditions on the torus.
We also give a proof of the H 1/2 Sobolev regularity result, due to Agoshkov, for a general class of measures considered in 9 . We use analogous arguments as in 9 but the result is not a consequence of 9, Theorem 4 , see the commentary on the beginning of Section 4 .
Finally, we prove how suitable assumptions on the abstract velocity measure signed for N > 1 can be useful to derive the H 2 Sobolev regularity for the operator M λ .
Before closing the introduction, let us recall that the smoothing effect of velocity averages M 0 , with k 1 1 , in terms of the H 1/2 Sobolev regularity, is given in 10 when the velocity space is endowed with the Lebesque measure, while a systematic analysis of the fractional Sobolev regularity, for general velocity measures in L p space p > 1 , is given in 9 see also 11-16 . These results play a cornerstone role in the analysis of kinetic models see, e.g., 17, 18 . Finally, velocity averages turn out to play an important role in the context of inverse problems see 19, 20 .

W
where dS is the Lebesgue measure on S N−1 the unit sphere of R N and dα is a positive bounded measure on 0, 1 such that for simplicity we assume that R 1

2.2
Let ψ ∈ L p dx ⊗ dμ v . It is easy to show that for any λ > −λ * To prove that N λ is an integral operator, we set

2.5
Since D is convex, the change of variables x x − tw dx t N−1 dt dS w leads to Thus, N λ is an integral operator with kernel Let us now introduce the following hypotheses: Moreover, N λ L L p dx⊗dμ v ;W 1,p x is locally bounded with respect to λ > −λ * .
Proof . Let us first compute ∂N λ ϕ/∂x i in the distributional sense for ϕ ∈ D D × V . To this end, let ψ ∈ D D × V , and extend k i ·, · , i 1, 2, by zero outside of V ; then

2.15
The use of the Green formula leads to

2.16
We claim that the second part of 2.16 goes to zero as ε → 0. Indeed,

2.17
Clearly Thus, using the dominated convergence theorem of Lebesgue, it is enough to prove that I 2 → 0 as ε goes to zero. This follows from the estimate which proves the claim.
Therefore, ∂N λ /∂x i may be decomposed as

2.22
Now, 2.16 becomes Clearly, Hence, according to A2 -A4 and the dominated convergence theorem of Lebesgue Therefore, using the Hölder inequality and the boundedness of D, the mapping Note, parenthetically, that from 2.25 we deduce the second part of the theorem. Next, to deal with J 1 , we first consider the truncated operator It is easily seen that S i ·, v, v is positively homogeneous of degree 0 and even and satisfies Furthermore, the truncated maximal operator S * i defined by see 3, 21 By the Hölder inequality and 2.29 where q is the conjugate of p, that is, q p/ p − 1 . Finally, by the dominated convergence theorem of Lebesque The density of D in L p dx ⊗ dμ concludes the proof.

in Periodic Transport
Let us first precise the functional setting of our problem. Let μ be a positive bounded measure on V satisfying the following: dμ is invariant by symmetry with respect to the origin 3.1 ess-sup where C is a positive constant.

ISRN Mathematical Analysis
We define the streaming operator T p on L 2 dx ⊗ dμ v by where D 0, 2π N . We expand ψ ∈ L 2 dx ⊗ dμ v into the Fourier series with respect to x: and by the Parseval formula By simple computations we have The regularity result of N λ is obtained under the following two weaker assumptions: where G is defined by 2.2 . The following lemma will play a crucial role in the proof of the main result.
where C 0 is a positive constant.
where α < 1 is to be chosen later. It is clear that Now, let β be the image of μ under the orthogonal projection on the direction k/|k|. Hence,

3.13
Let ζ t t α dβ s for t > α. An integration by parts yields By choosing α ε/|k| with ε < 1, we obtain where C 0 is a positive constant. This achieves the proof of the lemma.

ISRN Mathematical Analysis
Theorem 3.2. Under assumptions A7 -A8 , the operator N λ λ > −λ * maps continuously and thanks to the evenness of σ · , G ·, v, v together with the fact that dμ is invariant by symmetry with respect to the origin, the last integral vanishes. Therefore, 3.20 and consequently

3.21
Now, A8 and Lemma 3.1 lead to

3.22
By the Hölder inequality, we obtain This ends the proof of the theorem. Our notations and assumptions on dμ are the same as in the preceding section. Let k ·, · be a measurable function on V × V such that It is announced without proof, in 10 , that M λ maps continuously L 2 dx ⊗ dμ v into H 1/2 , where dμ is the Lebesgue measure and k ·, · 1. We propose a proof for this result see Theorem 4.1 for a class of measures dμ satisfying 3.2 . The proof is inspired from 9 but is not a consequence of the results of 9 because the traces of ψ ∈ D T p do not lie in L 2 ∂D × V ; |v · n x |dσ x dμ v , but in a certain greatest weighted L 2 space see 1, 2, 4 for details .
We define Theorem 4.1. Assume that k ·, · 1. Then, for any λ > −λ * , the operator M λ maps continuously Proof. Let ψ ∈ L 2 dx ⊗ dμ v . We expand ψ into the Fourier series: 4.4 We have dμ v e i x·k .

4.5
By the Hölder inequality and Lemma 3.1

4.6
The proof is achieved by the Parseval formula and the boundedness of 1 |k| 2 1/2 /|k|. Now, we are going to prove the H 1 regularity of M λ when restricted to a subspace of L 2 dx dμ consisting of even source term.

Theorem 4.3. Let A7 be satisfied. In addition, suppose that
Then, for any λ > −λ * , the operator M λ maps continuously O into H 1 Proof. We proceed as in the proof of Theorem 3.2. We expand ψ ∈ O into the Fourier series: 4.9 We have

4.10
According to A7 and A9 The use of Lemma 3.1 gives

4.12
Since sup v∈V |f k v | k∈Z N ∈ l 2 , we deduce from the Parseval formula that ∂/∂x j M λ ψ ∈ L 2 dx ⊗ dμ v , which concludes the proof.

Remark 4.4.
Arguing as in the proof of Lemma 3.1, one sees that where C 0 is a positive constant. So, by the Hölder inequality Theorem 4.3 is still true if we replace O by 4.14 which is of interest for neutron transport approximations. The subspace O can be also replaced by

H 2 Regularity of the Operator
In this section we focus our attention on the smoothing effect of the velocity averages H 2 under some particular assumptions on the abstract measure dμ. For technical reasons, we treat separately the cases N 1 and N > 1.

H 2 x of Regularity M λ in One Dimension
Let dα be a bounded measure not necessarily positive on −1, 1 satisfying the following assumptions: A10 dα is invariant by symmetry with respect to zero, 5.1 where d|α| is the absolute value of the measure dα,

5.5
Then, we have the following.

5.8
Note that the last integral of 5.8 vanishes because its integrand is odd, in view of assumptions A10 , A12 -A13 and the evenness of f k · . Now, using the Hölder inequality together with assumption A10 , we obtain

H 2 Regularity of
where dS w is the Lebesgue measure on S N−1 and dα is a signed measure on 0, 1 satisfying the following conditions: The more technical assumption A14 plays a key role for our analysis. In the sequel, we show the optimality of this assumption. We need also the following assumption:
Proof. Let ψ ∈ L 2 D . We use the Fourier series of ψ x k∈Z N f k e i x·k . We recall that where σ ρ λ σ ρ .
Let us recall 22 that for all fixed w 0 ∈ S N−1 Accordingly, dt e i x·k .

5.18
Set We give the proof for N > 3 the other cases are similar . An integration by parts with respect to t yields

5.23
Now, thanks to A14 , the last term of 5.23 vanishes. Set

5.27
we deduce from the Parseval formula that ∂/∂x i ∂/∂x j M λ ψ ∈ L 2 D . This achieves the proof.
Remark 5.3. 1 Expression 5.21 shows the optimality of assumption A14 because |k|f k k∈Z N is not necessary in l 2 . 2 Note that in 19, 20 the use of velocity averages in the context of inverse problems has been studied. The problems consist in the explicit determination of the spatial parts of internal sources from two suitable moments velocity averages of the solution of integrodifferential transport equations for classical vacuum boundary conditions see 19 or periodic boundary conditions see 20 by means of appropriate signed measures.