Abstract
The main purpose of this paper is to extend the regularity results in neutron transport theory, with respect to the Lebesgue measure due to Mokhtar-Kharroubi, (1991), 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. regularity results are derived when the velocity space is endowed with an appropriate class of measures (signed in the multidimensional case).
1. Introduction
Let be an open bounded convex subset of . Let be a positive bounded measure on supported by . We denote
where , , and is the Lebesgue measure on .
The streaming operator is defined where Here, is the unit outward normal at . It is well known that under the hypothesis , any function possesses traces on (see [1–4]).
Let be two measurable functions on such that
Note that, for all , the operators and map continuously into itself. In [5], several regularity results of the operators and were established when the velocity space 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 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 ) can be useful to derive the Sobolev regularity for the operator .
Before closing the introduction, let us recall that the smoothing effect of velocity averages (, with ), in terms of the 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 space , 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]).
2. Regularity of the Operator
Let , where is the Lebesgue measure on (the unit sphere of ) and is a positive bounded measure on such that (for simplicity we assume that ) For , we set Let . It is easy to show that for any where
To prove that is an integral operator, we set Since is convex, the change of variables () leads to Thus, is an integral operator with kernel Let us now introduce the following hypotheses: (
Remark 2.1. (1) is the key assumption.
(2) Assumption is a consequence of .
(3) Assumption is not necessary if is the Lebesgue measure (see [5]).
Throughout this paper we set
Now, we are ready to state our first main result.
Theorem 2.2. Let assumptions – be satisfied. Then, for any and , the operator maps continuously into , where Moreover, is locally bounded with respect to .
Proof . Let us first compute in the distributional sense for . To this end, let , and extend , , by zero outside of ; then
The use of the Green formula leads to
We claim that the second part of (2.16) goes to zero as .
Indeed,
Clearly (because ). Thus, using the dominated convergence theorem of Lebesgue, it is enough to prove that as goes to zero. This follows from the estimate
which proves the claim.
Next, we compute the derivative of the kernel .
Indeed, by straightforward calculations
where
It is easily seen that , where is a bounded function.
Therefore, may be decomposed as
Set
Now, (2.16) becomes
Clearly,
where
Hence, according to – and the dominated convergence theorem of Lebesgue
Therefore, using the Hölder inequality and the boundedness of , the mapping
defines a bounded operator .
Note, parenthetically, that from (2.25) we deduce the second part of the theorem.
Next, to deal with , we first consider the truncated operator
It is easily seen that is positively homogeneous of degree 0 and even and satisfies
So, is a Caldéron-Zygmund operator which converges in , as to (see [3, 21]), where
Furthermore, the truncated maximal operator defined by (see [3, 21])
satisfies
By the Hölder inequality and (2.29)
where is the conjugate of , that is, . Finally, by the dominated convergence theorem of Lebesque
which amounts to
The density of in concludes the proof.
Remark 2.3. We note that Theorem 2.2 has already been obtained by [5, Theorem 1] of Mokhtar-Kharroubi when the velocity space is endowed with the Lebesque measure. The choice of an abstract measure is motivated by our desire to give a unified treatment which covers either continuous models (the Lebesque measures on open subsets of ) or multigroup models (the Lebesque measure on spheres).
3. Regularity of the Operator in Periodic Transport
Let us first precise the functional setting of our problem. Let be a positive bounded measure on satisfying the following: where is a positive constant.
We define the streaming operator on by where . We expand into the Fourier series with respect to : where and by the Parseval formula By simple computations we have The regularity result of is obtained under the following two weaker assumptions: (
where is defined by (2.2). The following lemma will play a crucial role in the proof of the main result.
Lemma 3.1. Let such that . Then, where is a positive constant.
Proof. We write where is to be chosen later. It is clear that Now, let be the image of under the orthogonal projection on the direction . Hence, Let for . An integration by parts yields Since by (3.2), it follows that By choosing with , we obtain where is a positive constant. This achieves the proof of the lemma.
Theorem 3.2. Under assumptions –, the operator maps continuously into , where Moreover, is locally bounded with respect to .
Proof. Let . We recall that Since and thanks to the evenness of together with the fact that is invariant by symmetry with respect to the origin, the last integral vanishes. Therefore, and consequently Now, and Lemma 3.1 lead to By the Hölder inequality, we obtain and by the Parseval formula This ends the proof of the theorem.
Remark 3.3. The measure , with , satisfies (3.2). Thus, it covers the continuous models as well as the multigroup models in periodic Transport.
4. Regularity of the Operator
Our notations and assumptions on are the same as in the preceding section. Let be a measurable function on such that
It is announced without proof, in [10], that maps continuously into , where is the Lebesgue measure and . We propose a proof for this result (see Theorem 4.1) for a class of measures satisfying (3.2). The proof is inspired from [9] but is not a consequence of the results of [9] because the traces of do not lie in , but in a certain greatest weighted space (see [1, 2, 4] for details).
We define where
Theorem 4.1. Assume that . Then, for any , the operator maps continuously into .
Proof. Let . We expand into the Fourier series: We have By the Hölder inequality and Lemma 3.1 The proof is achieved by the Parseval formula and the boundedness of .
Remark 4.2. Assumption (3.1) is unnecessary in the proof of Theorem 4.1 but it plays a key role in the following theorem.
Now, we are going to prove the regularity of when restricted to a subspace of consisting of even source term.
Theorem 4.3. Let be satisfied. In addition, suppose that Then, for any , the operator maps continuously into , where
and .
Moreover, is locally bounded with respect to .
Proof. We proceed as in the proof of Theorem 3.2. We expand into the Fourier series: We have According to and The use of Lemma 3.1 gives Since , we deduce from the Parseval formula that , which concludes the proof.
Remark 4.4. Arguing as in the proof of Lemma 3.1, one sees that
where is a positive constant. So, by the Hölder inequality Theorem 4.3 is still true if we replace by
which is of interest for neutron transport approximations.
The subspace can be also replaced by
where .
5. Regularity of the Operator
In this section we focus our attention on the smoothing effect of the velocity averages () under some particular assumptions on the abstract measure . For technical reasons, we treat separately the cases and .
5.1. of Regularity in One Dimension
Let be a bounded measure (not necessarily positive) on satisfying the following assumptions:
where is the absolute value of the measure ,
Let be a measurable function such that
Let us introduce the following subspace of : Then, we have the following.
Theorem 5.1. Let assumptions – be satisfied. Then, for any operator the operator maps continuously into Moreover, is locally bounded with respect to .
Proof. For we have Thus, Note that the last integral of (5.8) vanishes because its integrand is odd, in view of assumptions – and the evenness of . Now, using the Hölder inequality together with assumption , we obtain Next, since and , it follows from the Parseval formula that where is locally bounded in , and the proof is complete.
5.2. Regularity of
Let , where is the Lebesgue measure on and is a signed measure on satisfying the following conditions:
The more technical assumption plays a key role for our analysis. In the sequel, we show the optimality of this assumption. We need also the following assumption:
Theorem 5.2. Let the hypotheses – be satisfied. Then, for any , the operator maps continuously into , where is the averaging operator.
Proof. Let . We use the Fourier series of . We recall that
where .
Let us recall [22] that for all fixed
Accordingly,
Set
We give the proof for (the other cases are similar).
An integration by parts with respect to yields
Thanks to , (5.20) becomes
Set
An integration by parts for the Stieltjes measures yields
Now, thanks to , the last term of (5.23) vanishes. Set
Accordingly, (5.20) becomes
Consequently,
Since
we deduce from the Parseval formula that . This achieves the proof.
Remark 5.3. (1) Expression (5.21) shows the optimality of assumption () because is not necessary in .
(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 integro-differential transport equations for classical vacuum boundary conditions (see [19]) or periodic boundary conditions (see [20]) by means of appropriate signed measures.
Acknowledgments
The author would like to thank Professor M. Mokhtar-Kharroubi for suggesting these problems and for many helpful comments. He also expresses his acknowledgment to the referees for their valuable suggestions that improved the original paper.