# Sobolev Regularity in Neutron Transport Theory

Authors: Ahmed Zeghal
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

The main purpose of this paper is to extend theW1,p 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. H2 regularity results are derived when the velocity space is endowed with an appropriate class of measures (signed in the multidimensional case).


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## Body

## 1. Introduction

LetD be an open bounded convex subset of ℝN. Let dμ be a positive bounded measure on ℝN supported by V={v∈ℝN;|v|<R},0<R<+∞. We denote (1.1)Xp={ψ∈Lp(dx⊗dμ(v));v⋅∂ψ∂x∈Lp(dx⊗dμ(v))},whereLp(dx⊗dμ(v))=Lp(D×V;dx⊗dμ(v)), 1≤p<+∞, and dx is the Lebesgue measure on D.The streaming operatorT is defined (1.2)Tψ=-v⋅∂ψ∂x-σ(v)ψ,D(T)={ψ∈Xp;ψ|Γ-=0},
where(1.3)σ(⋅)∈L∞(V;dμ(v)),Γ-={(x,v)∈∂D×V;v⋅n(x)<0}.
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∈Xp possesses traces on Γ- (see [1–4]).Letki(·,·)(i=1,2) be two measurable functions on V×V such that(1.4)Ki∈L(Lp(V;dμ(v))),whereKiis the operator with kernelki,i.e.Ki:φ∈Lp(V;dμ(v))⟶∫Vki(⋅,v′)φ(v′)dμ(v′).Note that, for allλ>-λ*=-dμess-infσ(·), the operators 𝒩λ=K1(λ-T)-1K2 and ℳλ=K1(λ-T)-1 map continuously Lp(dx⊗dμ(v)) into itself. In [5], several W1,p regularity results of the operators 𝒩λ and ℳλ 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 theH1/2 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 forN>1) can be useful to derive the H2 Sobolev regularity for the operator ℳλ.Before closing the introduction, let us recall that the smoothing effect of velocity averages (ℳ0, with k1=1), in terms of the H1/2 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 Lp 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]).


## 2.Wx1,p Regularity of the Operator 𝒩λ=K1(λ-T)-1K2

Letdμ(v)=dα(ρ)⊗dS(w), where dS is the Lebesgue measure on 𝕊N-1 (the unit sphere of ℝN) and dα is a positive bounded measure on [0,1) such that (for simplicity we assume that R=1)(2.1)∫01dα(ρ)ρ<∞.
For (z,v,v′)∈V, we set(2.2)G(z,v,v′):=k1(v,z)k2(z,v′),λ*:=dμess-inf⁡⁡σ(⋅).
Let ψ∈Lp(dx⊗dμ(v)). It is easy to show that for any λ>-λ*(2.3)Nλψ(x,v)=K1(λ-T)-1K2ψ(x,v)=∫Vdμ(v′)∫Vdμ(v′′)∫0t(x,v′′)G(v′′,v,v′)e-t(λ+σ(v′′))ψ(x-tv′′,v′)dt,
where (2.4)t(x,v)=inf⁡{t>0,x-tv∉D}.To prove that𝒩λ is an integral operator, we set(2.5)I=∫Vdμ(v′′)∫0t(x,v′′)G(v′′,v,v′)e-t(λ+σ(v′′))ψ(x-tv′′,v′)dt=∫01dα(ρ)∫SN-1dS(w)∫0t(x,w)/ρG(ρw,v,v′)e-t(λ+σ(ρw))ψ(x-tρw,v′)dt=∫01dα(ρ)ρ∫SN-1dS(w)∫0t(x,w)G(ρw,v,v′)e-(t/ρ)(λ+σ(ρw))ψ(x-tw,v′)dt.
Since D is convex, the change of variables x′=x-tw (dx′=tN-1dtdS(w)) leads to(2.6)I=∫01dα(ρ)ρ∫DG(ρ((x-x′)/|x-x′|),v,v′)e-(|x-x′|/ρ)(λ+σ(ρ((x-x′)/|x-x′|)))|x-x′|N-1ψ(x′,v′)dx′.
Thus, 𝒩λ is an integral operator with kernel(2.7)Nλ(x-x′,v,v′)=∫01G(ρ((x-x′)/|x-x′|),v,v′)e-(|x-x′|/ρ)(λ+σ(ρ((x-x′)/|x-x′|)))|x-x′|N-1dα(ρ)ρ.
Let us now introduce the following hypotheses: (𝒜1)
(2.8)σ(z)=σ(-z),dμ(v)a.e.,G(z,v,v′)=G(-z,v,v′),dμ(v)a.e.,(𝒜2)
(2.9)G∞∈L(Lp(V;dμ(v))),whereG∞is the integral operator with kernelG∞(v,v′):=sup⁡z∈V|G(z,v,v′)|,(𝒜3)
(2.10)G¯∈L(Lp(V;dμ(v))),whereG¯is the integral operator with kernelG¯(v,v′):=sup⁡z∈V|G(z,v,v′)||z|,(𝒜4)
(2.11)Gi∈L(Lp(V;dμ(v))),whereGiis the integral operator with kernelGi(v,v′):=sup⁡z∈V|∂G∂zi(z,v,v′)|,1≤i≤N,(𝒜5)
(2.12)σ(⋅)∈W1,∞(V;dμ(v)).Remark 2.1.
(1)(𝒜1) is the key assumption.
(2) Assumption(𝒜2) is a consequence of (𝒜3).
(3) Assumption(𝒜3) is not necessary if dμ is the Lebesgue measure (see [5]).
Throughout this paper we set(2.13)σ¯(⋅)=λ+σ(⋅).Now, we are ready to state our first main result.Theorem 2.2.
Let assumptions(𝒜1),(𝒜3)–(𝒜5) be satisfied. Then, for any λ>-λ* and p>1, the operator 𝒩λ maps continuously Lp(dx⊗dμ(v)) into Wx1,p(dx⊗dμ(v)), where
(2.14)Wx1,p(dx⊗dμ(v)):={ψ∈Lp(dx⊗dμ(v))such that∂ψ∂xi∈Lp(dx⊗dμ(v))}.
Moreover, ∥𝒩λ∥ℒ(Lp(dx⊗dμ(v));Wx1,p) is locally bounded with respect to λ>-λ*.Proof .
Let us first compute∂𝒩λφ/∂xi in the distributional sense for φ∈𝒟(D×V). To this end, let ψ∈𝒟(D×V), and extend ki(·,·), i=1,2, by zero outside of V; then
(2.15)〈∂Nλφ∂xi,ψ〉D′,D=-∫D×V∂ψ∂xi(x,v)dxdμ(v)∫D×VNλ(x-x′,v,v′)φ(x′,v′)dx′dμ(v′)=-lim⁡ɛ→0∫V×Vdμ(v)dμ(v′)∫RNφ(x′,v′)dx′×∫|x-x′|≥ϵNλ(x-x′,v,v′)∂ψ∂xi(x,v)dx.
The use of the Green formula leads to
(2.16)lim⁡ɛ→0[∫V×Vdμ(v)dμ(v′)∫RNφ(x′,v′)dx′∫|x-x′|≥ϵ∂Nλ∂xi(x-x′,v,v′)ψ(x,v)dx-∫V×Vdμ(v)dμ(v′)∫RNφ(x′,v′)dx′∫|x-x′|=ϵNλ(x-x′,v,v′)ψ(x,v)νidσ(x)].
We claim that the second part of (2.16) goes to zero asɛ→0.
Indeed,(2.17)∫|x-x′|=ϵNλ(x-x′,v,v′)ψ(x,v)νidσ(x)=ψ(x′,v)∫|x-x′|=ϵNλ(x-x′,v,v′)νidσ(x)+∫|x-x′|=ϵNλ(x-x′,v,v′)|x-x′|O(|x-x′|)νidσ(x)=I1+I2.
Clearly I1=0 (because Nλ(y,v,v′)=Nλ(-y,v,v′)). Thus, using the dominated convergence theorem of Lebesgue, it is enough to prove that I2→0 as ɛ goes to zero. This follows from the estimate
(2.18)|I2|≤(∫01dα(ρ)ρ)G∞(v,v′)∫|x-x′|=ɛdσ(x)ɛN-2=(∫01dα(ρ)ρ)G∞(v,v′)|SN-1|ɛ,
which proves the claim.
Next, we compute the derivative of the kernelNλ.
Indeed, by straightforward calculations(2.19)∂Nλ∂xi(x,v,v′)=-∑j≠i∫01xixj|x|N+2Gj(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)+∫01|x|2-xi2|x|N+2Gi(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)+∑j≠i∫01xixj|x|N+1σj(ρx|x|)G(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ-∫01|x|2-xi2|x|N+1σi(ρx|x|)G(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ-∫01xi|x|Nσ̅(ρx|x|)G(ρ(x/|x|),v,v′)ρe-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ-(N-1)∫01xi|x|N+1G(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ,
where
(2.20)σj(⋅):=∂σ∂zj(⋅),Gj(⋅,v,v′):=∂G∂zj(⋅,v,v′).
It is easily seen thate-(|x|/ρ)σ¯(ρ(x/|x|))=1-(|x|/ρ)σ¯(ρ(x/|x|))B(x,ρ), where B(·,·) is a bounded function.
Therefore,∂Nλ/∂xi  may be decomposed as
(2.21)∂Nλ∂xi(x,v,v′)=-∑j≠i∫01xixj|x|N+2Gj(ρx|x|,v,v′)dα(ρ)+∫01|x|2-xi2|x|N+2Gi(ρx|x|,v,v′)dα(ρ)+∑j≠i∫01xixj|x|N+1Gj(ρx|x|,v,v′)σ¯(ρx|x|)B(x,ρ)dα(ρ)ρ-∫01|x|2-xi2|x|N+1Gi(ρx|x|,v,v′)σ¯(ρx|x|)B(x,ρ)dα(ρ)ρ+∑j≠i∫01xixj|x|N+1σj(ρx|x|)G(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ-∫01|x|2-xi2|x|N+1σi(ρx|x|)G(ρx|x|,v,v′)e-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ-∫01xi|x|Nσ¯(ρx|x|)G(ρ(x/|x|),v,v′)ρe-(|x|/ρ)σ̅(ρ(x/|x|))dα(ρ)ρ-(N-1)∫01xi|x|N+1G(ρx|x|,v,v′)dα(ρ)ρ+(N-1)∫01xi|x|NG(ρ(x/|x|),v,v′)ρσ¯(ρx|x|)B(x,ρ)dα(ρ)ρ.
Set
(2.22)Si(x,v,v′):=-∑j≠i∫01xixj|x|2Gj(ρx|x|,v,v′)dα(ρ)+∫01|x|2-xi2|x|2Gi(ρx|x|,v,v′)dα(ρ)-(N-1)∫01xi|x|G(ρx|x|,v,v′)dα(ρ)ρ,Ri(x,v,v′):=∑j≠i∫01xixj|x|N+1Gj(ρx|x|,v,v′)σ¯(ρx|x|)B(x,ρ)dα(ρ)ρ-∫01|x|2-xi2|x|N+1Gi(ρx|x|,v,v′)σ¯(ρx|x|)B(x,ρ)dα(ρ)ρ+∑j≠i∫01xixj|x|N+1σj(ρx|x|)G(ρx|x|,v,v′)e-(|x|/ρ)σ¯(ρ(x/|x|)dα(ρ)ρ-∫01|x|2-xi2|x|N+1σi(ρx|x|)G(ρx|x|,v,v′)e-(|x|/ρ)σ¯(ρ(x/|x|)dα(ρ)ρ-∫01xi|x|Nσ¯(ρx|x|)G(ρ(x/|x|),v,v′)ρe-(|x|/ρ)σ¯(ρ(x/|x|))dα(ρ)ρ+(N-1)∫01xi|x|NG(ρ(x/|x|),v,v′)ρσ¯(ρx|x|)B(x,ρ)dα(ρ)ρ.
Now, (2.16) becomes
(2.23)〈∂Nλφ∂xi,ψ〉D′,D=lim⁡ɛ→0[∫V×Vdμ(v)dμ(v′)∫RNψ(x,v)dx∫|x-x′|≥ϵRi(x-x′,v,v′)φ(x′,v′)dx′+∫V×Vdμ(v)dμ(v′)∫RNψ(x,v)dx∫|x-x′|≥ϵSi(x-x′,v,v′)φ(x′,v′)|x-x′|Ndx′]=J1+J2.
Clearly,
(2.24)|Ri(x-x′,v,v′)|≤C(G∞(v,v′)+G¯(v,v′)+∑j=1NGj(v,v′))|x-x′|N-1,
where
(2.25)C=(∫01dα(ρ)ρ)max⁡(∑j=1N‖σj‖∞,(N‖B‖∞+1)(λ+‖σ‖∞)).
Hence, according to (𝒜2)–(𝒜4) and the dominated convergence theorem of Lebesgue
(2.26)J2=∫D×Vψ(x,v)dxdμ(v)∫D×VRi(x-x′,v,v′)φ(x′,v′)dx′dμ(v′).
Therefore, using the Hölder inequality and the boundedness of D, the mapping
(2.27)ψ⟶∫D×VRi(x-x′,v,v′)ψ(x′,v′)dx′dμ(v′)defines a bounded operator ℛi∈ℒ(Lp(dx⊗dμ)).
Note, parenthetically, that from (2.25) we deduce the second part of the theorem.
Next, to deal withJ1, we first consider the truncated operator (2.28)Si,ϵ:φ⟶∫|x-x′|≥ϵSi(x-x′,v,v′)φ(x′,v′)|x-x′|Ndx′for fixed(v,v′).
It is easily seen that Si(·,v,v′) is positively homogeneous of degree 0 and even and satisfies
(2.29)|Si(x,v,v′)|≤(∫01dα(ρ)ρ)((N-1)G∞(v,v′)+∑j=1NGj(v,v′)).
So, 𝒮i,ϵ is a Caldéron-Zygmund operator which converges in Lp, as ɛ→0 to 𝒮i (see [3, 21]), where
(2.30)Siφ:=p⋅v⋅∫RNSi(x-x′,v,v′)φ(x′,v′)|x-x′|Ndx′.
Furthermore, the truncated maximal operator 𝒮i* defined by (see [3, 21])
(2.31)Si*:=sup⁡ɛ>0|Si*φ|
satisfies
(2.32)∥Si*∥Lxp(RN)≤C′∥Si(⋅,v,v′)∥L∞(SN-1)∥φ(⋅,v′)∥Lxp(RN).
By the Hölder inequality and (2.29)
(2.33)|∫ψ(x,v)Si,ɛφ(x,v)dx|≤C′′[(N-1)G∞(v,v′)+∑j=1NGj(v,v′)](∫|ψ(x,v)|q)1/q∥φ(⋅,v′)∥Lxp(RN),
where q is the conjugate of p, that is, q=p/(p-1). Finally, by the dominated convergence theorem of Lebesque
(2.34)J2=(Siφ,ψ)Lp,Lq,Si∈L(Lp(RN×V;dx⊗dμ(v))),
which amounts to
(2.35)〈∂Nλφ∂xi,ψ〉D′,D=(Riφ,ψ)Lp,Lq+(Siφ,ψ)Lp,Lq.
The density of 𝒟 in Lp(dx⊗dμ) concludes the proof.Remark 2.3.
We note that Theorem2.2 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 ℝN) or multigroup models (the Lebesque measure on spheres).


## 3.Hx1 Regularity of the Operator 𝒩λ=K1(λ-Tp)-1K2 in  Periodic Transport

Let us first precise the functional setting of our problem. Letμ be a positive bounded measure on V satisfying the following:
(3.1)dμisinvariantbysymmetrywithrespecttotheorigin(3.2)ess-sup⁡w∈SN-1μ{v∈V;|v⋅w|≤ϵ}≤Cϵ,
where C is a positive constant.We define the streaming operatorTp on L2(dx⊗dμ(v)) by(3.3)Tpψ=-v⋅∂ψ∂x-σ(v)ψ,D(Tp)={ψ∈X2;ψ|xi=0=ψ|xi=2π(1≤i≤N)},
where D=(0,2π)N. We expand ψ∈L2(dx⊗dμ(v)) into the Fourier series with respect to x:(3.4)ψ(x,v)=∑k∈ZNfk(v)ei(x⋅k),
where (3.5)fk(v)=1(2π)n/2∫Dψ(x,v)e-i(x⋅k)dx∈L2(V;dμ(v)),k∈Zn,
and by the Parseval formula (3.6)∥ψ∥L2(dx⊗dμ(v))2=∑k∈ZN∫V|fk(v)|2dμ(v)<∞.
By simple computations we have(3.7)Nλψ(x,v)=∑k∈ZN[∫Vfk(v′)dμ(v′)∫VG(v′′,v,v′)σ¯(v′′)+i(v′′⋅k)dμ(v′′)]ei(x⋅k).
The regularity result of 𝒩λ is obtained under the following two weaker assumptions: (𝒜7)
(3.8)σ(v)=σ(-v),dμ(v)a.e.,σ∈L∞(V;dμ(v)),(𝒜8)
(3.9)G(z,v,v′)=G(-z,v,v′),dμ(v)a.e.,G∞(v,v′):=ess-sup⁡z∈V|G(z,v,v′)|∈L2(V×V;dμ⊗dμ),whereG is defined by (2.2). The following lemma will play a crucial role in the proof of the main result.Lemma 3.1.
Letk∈ℤN such that k≠(0,…,0). Then,
(3.10)∫Vdμ(v)(λ+λ*)2+(v⋅k)2≤C0|k|,
where C0 is a positive constant.Proof.
We write(3.11)∫Vdμ(v)(λ+λ*)2+(v⋅k)2=∫|v⋅(k/|k|)|<αdμ(v)(λ+λ*)2+(v⋅k)2+∫|v⋅(k/|k|)|≥αdμ(v)(λ+λ*)2+(v⋅k)2,
where α<1 is to be chosen later. It is clear that
(3.12)∫|v⋅(k/|k|)|<αdμ(v)(λ+λ*)2+(v⋅k)2≤1(λ+λ*)2μ{v∈V,|v⋅k|k||<α}≤Cα(λ+λ*)2.
Now, let β be the image of μ under the orthogonal projection on the direction k/|k|. Hence,
(3.13)∫|v⋅(k/|k|)|≥αdμ(v)(λ+λ*)2+(v⋅k)2=∫α1dβ(t)(λ+λ*)2+|k|2t2=1|k|2∫α1|k|2t2(λ+λ*)2+|k|2t2dβ(t)t2≤1|k|2∫α1dβ(t)t2.
Let ζ(t)=∫αtdβ(s) for t>α. An integration by parts yields
(3.14)∫α1dβ(t)t2=[ζ(t)t2]α1+2∫α1ζ(t)t3dt.
Since ζ(t)=β([α,t])≤β([-t,t])≤Ct by (3.2), it follows that
(3.15)∫α1dβ(t)t2≤C+2Cα-2C≤2Cα.
By choosing α=ɛ/|k| with ɛ<1, we obtain
(3.16)∫Vdμ(v)(λ+λ*)2+(v⋅k)2≤C0|k|,
where C0 is a positive constant. This achieves the proof of the lemma.Theorem 3.2.
Under assumptions(𝒜7)–(𝒜8), the operator 𝒩λ(λ>-λ*) maps continuously L2(dx⊗dμ(v)) into Hx1(dx⊗dμ(v)), where
(3.17)Hx1(dx⊗dμ(v))={ψ∈L2(dx⊗dμ(v)),∂ψ∂xi∈L2(dx⊗dμ(v)),for1≤i≤N}.
Moreover, ∥𝒩λ∥ℒ(L2(dx⊗dμ(v));Hx1(dx⊗dμ(v))) is locally bounded with respect to λ>-λ*.Proof.
Letψ∈L2(dx⊗dμ(v)). We recall that
(3.18)Nλψ(x,v)=∑k∈ZN[∫Vfk(v′)dμ(v′)∫VG(v′′,v,v′)σ¯(v′′)+i(v′′⋅k)dμ(v′′)]ei(x⋅k).
Since
(3.19)∫VG(v′′,v,v′)σ¯(v′′)+i(v′′⋅k)dμ(v′′)=∫Vσ¯(v′′)G(v′′,v,v′)σ¯(v′′)2+(v′′⋅k)2dμ(v′′)-i∫V(v′′⋅k)G(v′′,v,v′)σ¯(v′′)2+(v′′⋅k)2dμ(v′′),
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)Nλψ(x,v)=∑k∈ZN[∫Vfk(v′)dμ(v′)∫Vσ¯(v′′)G(v′′,v,v′)σ¯(v′′)2+(v′′⋅k)2dμ(v′′)]ei(x⋅k),
and consequently
(3.21)∂∂xjNλψ(x,v)=∑k∈ZNikj[∫Vfk(v′)dμ(v′)∫Vσ¯(v′′)G(v′′,v,v′)σ¯(v′′)2+(v′′⋅k)2dμ(v′′)]ei(x⋅k).
Now, (𝒜8) and Lemma 3.1 lead to
(3.22)|∫Vσ¯(v′′)G(v′′,v,v′)σ¯(v′′)2+(v′′⋅k)2dμ(v′′)|≤(λ+‖σ‖∞)G∞(v,v′)∫Vdμ(v)(λ+λ*)2+(v⋅k)2≤C0(λ+‖σ‖∞)G∞(v,v′)|k|.
By the Hölder inequality, we obtain
(3.23)|kj∫Vfk(v′)dμ(v′)∫Vσ¯(v′′)G(v′′,v,v′)σ¯(v′′)2+(v′′⋅k)2dμ(v′′)|≤C0(λ+‖σ‖∞)(∫V|fk(v′)|2dμ(v′))1/2(∫VG∞(v,v′)2dμ(v′))1/2
and by the Parseval formula
(3.24)‖∂∂xiNλψ‖L2(dx⊗dμ(v))≤C0(λ+∥σ∥∞)∥G∞∥L2(V×V)∥ψ∥L2(dx⊗dμ(v)).
This ends the proof of the theorem.Remark 3.3.
The measuredμ(v)=dα(ρ)⊗dS(w), with ∫01(dα(ρ)/ρ)<+∞, satisfies (3.2). Thus, it covers the continuous models as well as the multigroup models in periodic Transport.


## 4.H1 Regularity of the Operator ℳλ=K(λ-Tp)-1

Our notations and assumptions ondμ are the same as in the preceding section. Let k(·,·) be a measurable function on V×V such that (4.1)K:φ⟶∫Vk(⋅,v′)φ(v′)dμ(v′)∈L(L2(V;dμ(v))).It is announced without proof, in [10], that ℳλ maps continuously L2(dx⊗dμ(v)) into H1/2, 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(Tp) do not lie in L2(∂D×V;|v·n(x)|dσ(x)dμ(v)), but in a certain greatest weighted L2 space (see [1, 2, 4] for details).We define(4.2)∥φ∥1/2=(2π)N/2[∑k∈ZN|fk|2(1+|k|2)1/2]1/2,
where (4.3)φ(x)=∑k∈ZNfkei(x⋅k)∈L2(D).Theorem 4.1.
Assume thatk(·,·)=1. Then, for any λ>-λ*, the operator ℳλ maps continuously L2(dx⊗dμ(v)) into H1/2(D).Proof.
Letψ∈L2(dx⊗dμ(v)). We expand ψ into the Fourier series:
(4.4)ψ(x,v)=∑k∈ZNfk(v)ei(x⋅k).
We have
(4.5)K(λ-Tp)-1ψ(x,v)=∑k∈ZN(∫Vfk(v)σ¯(v)+i(v⋅k)dμ(v))ei(x⋅k).
By the Hölder inequality and Lemma 3.1(4.6)(1+|k|2)1/2|∫Vfk(v)σ¯(v)+i(v⋅k)dμ(v)|2≤(1+|k|2)1/2(∫V|fk(v)|2dμ(v))∫Vdμ(v)(λ+λ*)2+(v⋅k)2≤C0(1+|k|2)1/2|k|∫V|fk(v)|2dμ(v).
The proof is achieved by the Parseval formula and the boundedness of (1+|k|2)1/2/|k|.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 theH1 regularity of ℳλ when restricted to a subspace of L2(dx⨂dμ) consisting of even source term.Theorem 4.3.
Let(𝒜7) be satisfied. In addition, suppose that (𝒜9)
(4.7)k(v,v′)=k(v,-v′),dμ(v)a.e.,ess-sup⁡v′∈V|k(⋅,v′)|=k∞(⋅)∈L2(V;dμ(v)).Then, for any λ>-λ*, the operator ℳλ maps continuously 𝒪 into Hx1(dx⊗dμ(v)), where
(4.8)O={ψ∈L2(dx⊗dμ(v)),ψ(x,v)=ψ(x,-v),(sup⁡v∈V|fk(v)|)k∈ZN∈l2}
and fk(·)=(1/(2π)N)∫Dψ(x,·)e-i(x·k)dx.
Moreover,∥ℳλ∥ℒ(L2(dx⊗dμ(v));Hx1(dx⊗dμ(v))) is locally bounded with respect to λ>-λ*.Proof.
We proceed as in the proof of Theorem3.2. We expand ψ∈𝒪 into the Fourier series:
(4.9)ψ(x,v)=∑k∈ZNfk(v)ei(x⋅k).
We have
(4.10)K(λ-Tp)-1ψ(x,v)=∑k∈ZN(∫Vk(v,v′)fk(v′)σ¯(v′)+i(v′⋅k)dμ(v′))ei(x⋅k).
According to (𝒜7) and (𝒜9)(4.11)∂∂xjMλψ(x,v)=∑k∈ZNikj(∫Vσ¯(v′)k(v,v′)fk(v′)σ¯(v′)2+(v′⋅k)2dμ(v′))ei(x⋅k).
The use of Lemma 3.1 gives
(4.12)|kj∫Vσ¯(v′)k(v,v′)fk(v′)σ¯(v′)2+(v′⋅k)2dμ(v′)|≤|kj|(λ+∥σ∥∞)k∞(v)(sup⁡v∈V|fk(v)|)×(∫Vdμ(v′)(λ+λ*)2+(v′⋅k)2)≤C0(λ+∥σ∥∞)k∞(v)sup⁡v∈V|fk(v)|.
Since (supv∈V|fk(v)|)k∈ℤN∈l2, we deduce from the Parseval formula that (∂/∂xj)ℳλψ∈L2(dx⊗dμ(v)), which concludes the proof.Remark 4.4.
Arguing as in the proof of Lemma3.1, one sees that
(4.13)∫Vdμ(v′)[(λ+λ*)2+(v′⋅k)2]2≤C0̂|k|,
where C0̂ is a positive constant. So, by the Hölder inequality Theorem 4.3 is still true if we replace 𝒪 by
(4.14){ψ(x,v)=∑j∈J,Jis finiteψj(x)φj(v),ψj∈L2(dx),φj∈L∞(dμ)andφj(v)=φj(-v)dμa.e.}.
which is of interest for neutron transport approximations.
The subspace𝒪 can be also replaced by
(4.15){ψ∈L2(dx⊗dμ(v)),ψ(x,v)=ψ(x,-v)and∑k∈ZN(1+|k|2)1/2∫v∈V|fk(v)|2dμ<∞},
where fk(·)=(1/2πN)∫Dψ(x,·)e-i(x·k)dx.


## 5.H2 Regularity of the Operator ℳλ=K(λ-Tp)-1

In this section we focus our attention on the smoothing effect of the velocity averages (H2) under some particular assumptions on the abstract measure dμ. For technical reasons, we treat separately the cases N=1 and N>1.


### 5.1.Hx2 of Regularity ℳλ in One Dimension

Letdα be a bounded measure (not necessarily positive) on (-1,1) satisfying the following assumptions: (𝒜10)
(5.1)dαis invariant by symmetry with respect to zero,(𝒜11)
(5.2)∫01d|α|(μ)μ2<∞,whered|α| is the absolute value of the measure dα, (𝒜12)
(5.3)σ∈L∞((-1,1);d|α|),σisevend|α|a.e.Letk(·,·) be a d|α| measurable function such that (𝒜13)
(5.4)k(⋅,μ′)=k(⋅,-μ′)d|α|a.e.,d|α|ess-sup⁡μ′∈(-1,1)|k(⋅,μ′)|=k∞(⋅)∈L2((-1,1);d|α|).Let us introduce the following subspace ofL2((0,2π)×(-1,1);dx⊗(d|α|/μ2)): (5.5)Ô={ψ∈L2((0,2π)×(-1,1);dx⊗d|α|μ2),ψ(⋅,μ)=ψ(⋅,-μ)d|α|a.e.}.
Then, we have the following.Theorem 5.1.
Let assumptions(𝒜10)–(𝒜13) be satisfied. Then, for any operator λ>-λ*=-d|α|ess-infσ(·) the operator ℳλ maps continuously 𝒪̂ into
(5.6)Hx2(dx⊗d|α|)={ψ∈L2(dx⊗d|α|),∂2ψ∂x2∈L2(dx⊗d|α|)}.
Moreover, ∥ℳλ∥ℒ(L2(dx⊗d|α|(μ)/μ2));Hx2(dx⊗d|α|)) is locally bounded with respect to λ>-λ*.Proof.
Forψ∈𝒪̂ we have
(5.7)K(λ-Tp)-1ψ(x,v)=∑k∈Z(∫-11k(μ,μ′)fk(μ′)λ+σ(μ′)+iμ′kdα(μ′))eixk.
Thus,
(5.8)∂2Mλψ∂x2(x,v)=-∑k∈Zk2(∫-11(λ+σ(μ′))k(μ,μ′)fk(μ′)λ+σ(μ′)+iμ′kdα(μ′))eixk=-∑k∈Zk2(∫-11(λ+σ(μ′))k(μ,μ′)fk(μ′)(λ+σ(μ′))2+μ′2k2dα(μ′)+ik∫-11μ′k(μ,μ′)fk(μ′)(λ+σ(μ′))2+μ′2k2dα(μ′))eixk.
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 fk(·). Now, using the Hölder inequality together with assumption (𝒜10), we obtain
(5.9)k2|∫-11k(μ,μ′)fk(μ′)λ+σ(μ′)+iμ′kdα(μ′)|≤2(λ+∥σ∥L∞(d|α|))k∞(μ)∫01k2μ2(λ+σ(μ))2+μ2k2|fk(μ)|d|α|(μ)μ2≤2(λ+∥σ∥L∞(d|α|))k∞(μ)(∫01d|α|(μ)μ2)1/2(∫01|fk(μ)|2d|α|(μ)μ2)1/2.
Next, since k∞(·)∈L2((-1,1);d|α|) and ∑k∈ℤ∫01|fk(μ)|2(d|α|(μ)/μ2)<∞, it follows from the Parseval formula that
(5.10)‖∂2Mλψ∂x2‖L2(dx⊗d|α|)≤C∥ψ∥L2(dx⊗(d|α|/μ2)),
where
(5.11)C=2(λ+∥σ∥L∞(d|α|))∥k∞∥L2(d|α|)(∫01d|α|(μ)μ2)1/2
is locally bounded in λ>-λ*, and the proof is complete.


### 5.2.H2 Regularity of ℳλ=M(λ-Tp)-1(N>1)

Letdμ(v)=dα(ρ)⊗dS(w), where dS(w) is the Lebesgue measure on 𝕊N-1 and dα is a signed measure on [0,1] satisfying the following conditions: (𝒜14)
(5.12)∫01dα(ρ)ρ=0,(𝒜15)
(5.13)∫01d|α|(ρ)ρ2<∞.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: (𝒜16)
(5.14)σ(v)=σ(|v|),σ(⋅)∈W1,∞([0,1]).Theorem 5.2.
Let the hypotheses(𝒜14)–(𝒜16) be satisfied. Then, for any λ>-λ*=-d|α|ess-infσ(·), the operator ℳλ=M(λ-Tp)-1 maps continuously L2(D) into H2(D), where
(5.15)M:φ∈L2(dx⊗dμ)⟶∫Vφ(x,v)dμ(v)∈L2(D),
is the averaging operator.Proof.
Letψ∈L2(D). We use the Fourier series of ψ(x)=∑k∈ℤNfkei(x·k). We recall that
(5.16)Mλψ(x,v)=∑k∈ZNfk∫01dα(ρ)∫SN-1dS(w)σ¯(ρ)+iρ(w⋅k)ei(x⋅k),
where σ¯(ρ)=λ+σ(ρ).
Let us recall [22] that for all fixed w0∈𝒮N-1(5.17)∫SN-1f(w⋅w0)dS(w)=|SN-2|∫-11f(t)(1-t2)(N-3)/2dt.
Accordingly,
(5.18)Mλψ(x,v)=f0|SN-1|∫01dα(ρ)σ¯(ρ)+2|SN-2|∑k≠0fk(∫01dα(ρ)∫01(1-t2)(N-3)/2σ¯(ρ)σ¯(ρ)2+ρ2|k|2t2dt)ei(x⋅k).
Set
(5.19)F0=f0|SN-1|∫01dα(ρ)σ¯(ρ).
We give the proof for N>3 (the other cases are similar).
An integration by parts with respect tot yields
(5.20)Mλψ(x,v)=F0+2(N-3)|SN-2|×∑k≠0fk|k|(∫01t(1-t2)(N-5)/2dt∫01arctan⁡(ρ|k|tσ¯(ρ))dα(ρ)ρ)ei(x⋅k).
Thanks to (𝒜14), (5.20) becomes
(5.21)Mλψ(x,v)=F0-2(N-3)|SN-2|×∑k≠0fk|k|(∫01t(1-t2)(N-5)/2dt∫01arctan(σ¯(ρ)ρ|k|t)dα(ρ)ρ)ei(x⋅k)+2π(N-3)|SN-2|∫01dα(ρ)ρ∫01t(1-t2)(N-5)/2dt∑k≠0fk|k|ei(x⋅k)=F0-2(N-3)|SN-2|×∑k≠0fk|k|(∫01t(1-t2)(N-5)/2dt∫01arctan(σ¯(ρ)ρ|k|t)dα(ρ)ρ)ei(x⋅k).
Set
(5.22)P(ρ)=∫0ρdα(s)s.
An integration by parts for the Stieltjes measures yields
(5.23)∫01arctan(σ¯(ρ)ρ|k|t)dP(ρ)=-1|k|t∫01P(ρ)(σ¯(ρ)ρ)′ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ+[P(ρ)arctan(σ¯(ρ)ρ|k|t)]ρ=0ρ=1.
Now, thanks to (𝒜14), the last term of (5.23) vanishes. Set
(5.24)Z(ρ)=P(ρ)(σ¯(ρ)ρ)′.
Accordingly, (5.20) becomes
(5.25)Mλψ(x,v)=F0+2(N-3)|SN-2|∑k≠0fk|k|2ei(x⋅k)×(∫01(1-t2)(N-5)/2dt∫01Z(ρ)ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ).
Consequently,
(5.26)∂∂xi∂∂xj(Mλψ(x,v))=-2(N-3)|SN-2|∑k≠0kikj|k|2fkei(x⋅k)×(∫01(1-t2)(N-5)/2dt∫01Z(ρ)ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ).
Since
(5.27)2(N-3)|SN-2||kikj|k|2∫01(1-t2)(N-5)/2dt∫01Z(ρ)ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ|≤(N-2)|SN-2|(λ+‖σ‖∞)(∫01d|α|(ρ)ρ2),
we deduce from the Parseval formula that (∂/∂xi)(∂/∂xj)(ℳλψ)∈L2(D). This achieves the proof.Remark 5.3.
(1) Expression (5.21) shows the optimality of assumption (𝒜14) because (|k|fk)k∈ℤN is not necessary in l2.
(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.


## 5.1.Hx2 of Regularity ℳλ in One Dimension

Letdα be a bounded measure (not necessarily positive) on (-1,1) satisfying the following assumptions: (𝒜10)
(5.1)dαis invariant by symmetry with respect to zero,(𝒜11)
(5.2)∫01d|α|(μ)μ2<∞,whered|α| is the absolute value of the measure dα, (𝒜12)
(5.3)σ∈L∞((-1,1);d|α|),σisevend|α|a.e.Letk(·,·) be a d|α| measurable function such that (𝒜13)
(5.4)k(⋅,μ′)=k(⋅,-μ′)d|α|a.e.,d|α|ess-sup⁡μ′∈(-1,1)|k(⋅,μ′)|=k∞(⋅)∈L2((-1,1);d|α|).Let us introduce the following subspace ofL2((0,2π)×(-1,1);dx⊗(d|α|/μ2)): (5.5)Ô={ψ∈L2((0,2π)×(-1,1);dx⊗d|α|μ2),ψ(⋅,μ)=ψ(⋅,-μ)d|α|a.e.}.
Then, we have the following.Theorem 5.1.
Let assumptions(𝒜10)–(𝒜13) be satisfied. Then, for any operator λ>-λ*=-d|α|ess-infσ(·) the operator ℳλ maps continuously 𝒪̂ into
(5.6)Hx2(dx⊗d|α|)={ψ∈L2(dx⊗d|α|),∂2ψ∂x2∈L2(dx⊗d|α|)}.
Moreover, ∥ℳλ∥ℒ(L2(dx⊗d|α|(μ)/μ2));Hx2(dx⊗d|α|)) is locally bounded with respect to λ>-λ*.Proof.
Forψ∈𝒪̂ we have
(5.7)K(λ-Tp)-1ψ(x,v)=∑k∈Z(∫-11k(μ,μ′)fk(μ′)λ+σ(μ′)+iμ′kdα(μ′))eixk.
Thus,
(5.8)∂2Mλψ∂x2(x,v)=-∑k∈Zk2(∫-11(λ+σ(μ′))k(μ,μ′)fk(μ′)λ+σ(μ′)+iμ′kdα(μ′))eixk=-∑k∈Zk2(∫-11(λ+σ(μ′))k(μ,μ′)fk(μ′)(λ+σ(μ′))2+μ′2k2dα(μ′)+ik∫-11μ′k(μ,μ′)fk(μ′)(λ+σ(μ′))2+μ′2k2dα(μ′))eixk.
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 fk(·). Now, using the Hölder inequality together with assumption (𝒜10), we obtain
(5.9)k2|∫-11k(μ,μ′)fk(μ′)λ+σ(μ′)+iμ′kdα(μ′)|≤2(λ+∥σ∥L∞(d|α|))k∞(μ)∫01k2μ2(λ+σ(μ))2+μ2k2|fk(μ)|d|α|(μ)μ2≤2(λ+∥σ∥L∞(d|α|))k∞(μ)(∫01d|α|(μ)μ2)1/2(∫01|fk(μ)|2d|α|(μ)μ2)1/2.
Next, since k∞(·)∈L2((-1,1);d|α|) and ∑k∈ℤ∫01|fk(μ)|2(d|α|(μ)/μ2)<∞, it follows from the Parseval formula that
(5.10)‖∂2Mλψ∂x2‖L2(dx⊗d|α|)≤C∥ψ∥L2(dx⊗(d|α|/μ2)),
where
(5.11)C=2(λ+∥σ∥L∞(d|α|))∥k∞∥L2(d|α|)(∫01d|α|(μ)μ2)1/2
is locally bounded in λ>-λ*, and the proof is complete.


## 5.2.H2 Regularity of ℳλ=M(λ-Tp)-1(N>1)

Letdμ(v)=dα(ρ)⊗dS(w), where dS(w) is the Lebesgue measure on 𝕊N-1 and dα is a signed measure on [0,1] satisfying the following conditions: (𝒜14)
(5.12)∫01dα(ρ)ρ=0,(𝒜15)
(5.13)∫01d|α|(ρ)ρ2<∞.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: (𝒜16)
(5.14)σ(v)=σ(|v|),σ(⋅)∈W1,∞([0,1]).Theorem 5.2.
Let the hypotheses(𝒜14)–(𝒜16) be satisfied. Then, for any λ>-λ*=-d|α|ess-infσ(·), the operator ℳλ=M(λ-Tp)-1 maps continuously L2(D) into H2(D), where
(5.15)M:φ∈L2(dx⊗dμ)⟶∫Vφ(x,v)dμ(v)∈L2(D),
is the averaging operator.Proof.
Letψ∈L2(D). We use the Fourier series of ψ(x)=∑k∈ℤNfkei(x·k). We recall that
(5.16)Mλψ(x,v)=∑k∈ZNfk∫01dα(ρ)∫SN-1dS(w)σ¯(ρ)+iρ(w⋅k)ei(x⋅k),
where σ¯(ρ)=λ+σ(ρ).
Let us recall [22] that for all fixed w0∈𝒮N-1(5.17)∫SN-1f(w⋅w0)dS(w)=|SN-2|∫-11f(t)(1-t2)(N-3)/2dt.
Accordingly,
(5.18)Mλψ(x,v)=f0|SN-1|∫01dα(ρ)σ¯(ρ)+2|SN-2|∑k≠0fk(∫01dα(ρ)∫01(1-t2)(N-3)/2σ¯(ρ)σ¯(ρ)2+ρ2|k|2t2dt)ei(x⋅k).
Set
(5.19)F0=f0|SN-1|∫01dα(ρ)σ¯(ρ).
We give the proof for N>3 (the other cases are similar).
An integration by parts with respect tot yields
(5.20)Mλψ(x,v)=F0+2(N-3)|SN-2|×∑k≠0fk|k|(∫01t(1-t2)(N-5)/2dt∫01arctan⁡(ρ|k|tσ¯(ρ))dα(ρ)ρ)ei(x⋅k).
Thanks to (𝒜14), (5.20) becomes
(5.21)Mλψ(x,v)=F0-2(N-3)|SN-2|×∑k≠0fk|k|(∫01t(1-t2)(N-5)/2dt∫01arctan(σ¯(ρ)ρ|k|t)dα(ρ)ρ)ei(x⋅k)+2π(N-3)|SN-2|∫01dα(ρ)ρ∫01t(1-t2)(N-5)/2dt∑k≠0fk|k|ei(x⋅k)=F0-2(N-3)|SN-2|×∑k≠0fk|k|(∫01t(1-t2)(N-5)/2dt∫01arctan(σ¯(ρ)ρ|k|t)dα(ρ)ρ)ei(x⋅k).
Set
(5.22)P(ρ)=∫0ρdα(s)s.
An integration by parts for the Stieltjes measures yields
(5.23)∫01arctan(σ¯(ρ)ρ|k|t)dP(ρ)=-1|k|t∫01P(ρ)(σ¯(ρ)ρ)′ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ+[P(ρ)arctan(σ¯(ρ)ρ|k|t)]ρ=0ρ=1.
Now, thanks to (𝒜14), the last term of (5.23) vanishes. Set
(5.24)Z(ρ)=P(ρ)(σ¯(ρ)ρ)′.
Accordingly, (5.20) becomes
(5.25)Mλψ(x,v)=F0+2(N-3)|SN-2|∑k≠0fk|k|2ei(x⋅k)×(∫01(1-t2)(N-5)/2dt∫01Z(ρ)ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ).
Consequently,
(5.26)∂∂xi∂∂xj(Mλψ(x,v))=-2(N-3)|SN-2|∑k≠0kikj|k|2fkei(x⋅k)×(∫01(1-t2)(N-5)/2dt∫01Z(ρ)ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ).
Since
(5.27)2(N-3)|SN-2||kikj|k|2∫01(1-t2)(N-5)/2dt∫01Z(ρ)ρ2|k|2t2σ¯(ρ)2+ρ2|k|2t2dρ|≤(N-2)|SN-2|(λ+‖σ‖∞)(∫01d|α|(ρ)ρ2),
we deduce from the Parseval formula that (∂/∂xi)(∂/∂xj)(ℳλψ)∈L2(D). This achieves the proof.Remark 5.3.
(1) Expression (5.21) shows the optimality of assumption (𝒜14) because (|k|fk)k∈ℤN is not necessary in l2.
(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.



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*Source: 379491-2012-01-26.xml*