1. Introduction and Notations
The Boltzmann equation (1872) is an integrodifferential equation of the kinetic theory which is devoted to the study of evolutionary behavior of the gas in the one particle phase space of position and velocity. The time evolution of the state of a gas which is contained in a vessel D bounded by solid walls is determined on one hand by the behavior of the gas molecules at collisions with each other and on the other hand by the influence of the walls as well as by external forces; in the case where there are no external forces, this state is described by a scalar function f(x,v,t) which models the density function of gas particles having position x∈D and velocity v∈R3 at time t∈R. The integral of this function ∫∫D×R3fx,v,tdx dv gives the expectation value (statistical average) of the total mass of gas contained in the phase space D×R3. Under some assumptions, function f must satisfy the Boltzmann equation⋆∂f∂tx,v,t=-v·∇xfx,v,t+Jfx,·,tvcompleted by boundary and initial conditions. The first term in ⋆ is called streaming operator which is responsible for the motion of the particles between collisions, while the second one J(f(x,·,t)), which is bilinear, describes the mechanism of collisions. A solution to the initial boundary value problem for ⋆ and a proof of H-theorem are given by treating it under its abstract form (for more details, see [1]).
This equation is applied also to the transport of photons involved in studies of nuclear reactors, including calculations on the protection against radiation and calculations of warm-up of materials. The quantum behavior of neutrons occurs in collisions with nuclei, but for physicists these events of collisions can be considered as one-time events and instantaneous, which only the consequences are interested in. According to the energy of the incident neutron and the nucleus with which it interacts, different types of reactions can occur. The neutron can be absorbed or broadcasted or it causes the fission of the nucleus. Each reaction is characterized by the microscopic cross section. Between collisions, neutrons behave as classical particles, described by their position and speed. Uncharged (neutral particles), they move in a straight line at least for short distances for which we neglect the effect of the gravitation. The neutronic equations are naturally linear. Indeed, the neutron-neutron interactions can be neglected vis-a-vis neutron-matter interactions. The relationship between the neutron density and the density of the propagation medium (water, uranium oxyde,…) is of the order 10-15, which justifies this approximation. This assumption leads to simplifying the nonlinear version of the Boltzmann equation used in the kinetic theory of gases.
Without delayed neutrons, these equations can be written under the form(1)∂ψ∂tx,v,t+v·∇xψx,v,t-σvψx,v,t+∫Vκx,v,v′ψx,v′,tdμv′=0with initial data ψ(x,v,0)=ψ0(x,v), where (x,v)∈D×V. D is a smooth open subset of Rn and μ(·) is a positive Radon measure on Rn such that μ({0})=0 and V (admissible velocity space) denotes the support of μ. The function ψ(x;v;t) describes the distribution of the neutrons in a nuclear reactor occupying the region D. The functions σ(·) and κ(·,·,·) are called, respectively, the collision frequency and the scattering kernel.
Here, the boundary conditions which represent the interaction between the particles and ambient medium are given by a boundary bounded operator H satisfying (2)ψ-=Hψ+,where ψ- (resp., ψ+) is the restriction of ψ to Γ- (resp., Γ+) with Γ- (resp., Γ+) being the incoming (resp., outcoming) part of the phase space boundary and H is a linear bounded operator from a suitable function space on Γ+ to a similar one on Γ-. The classical boundary conditions (vacuum boundary, specular reflections, diffuse reflections, and periodic and mixed type boundary conditions) are special examples of our framework.
Let (x,v)∈D¯×V. We define the positive real numbers t±(x;v) by(3)t±x,v=supt>0; x±sv∈D, ∀0<s<t.Physically, t±(x,v) is the time taken by a neutron initially in x∈D with animated speed ±v to achieve (for the first time) the boundary of D.
We denote by Γ± the set(4)Γ±=x,v∈∂D×V; ±v·nx≥0,where nx is the outer unit normal vector at x∈∂D.
Let 1≤p<∞; we introduce the functional spaces (5)Wp=ψ∈Xp such that v·∇xψ∈Xp,where(6)Xp≔LpD×V;dxdμv.The spaces of traces are Lp±≔Lp(Γ±;|v·nx|dγ(x)dμ(v)). Here dγ(·) is the Lebesgue measure on ∂D.
Recall that, for every ψ∈Wp, we can define the traces ψ± on Γ±; unfortunately, these traces do not belong to Lp±. The traces lie only in Lp,loc± or precisely in a certain weighted Lp space (see [2–4], for details).
Define (7)Wp~=ψ∈Wp; ψ±∈Lp±.In this case H∈L(Lp+,Lp-) (1≤p<∞) and the associated advection operator TH is given as follows: (8)TH:DTH⊆Xp⟶Xp,φ⟶THφ=-v·∇xφx,v-σvφx,v,with domain (9)DTH=ψ∈Wp~ such that ψ-=Hψ+,where the collision frequency σ(·)∈L+∞(V) (in other words, a positive bounded function).
Let λ∈C; consider the boundary value problem(10)λψx,v+v·∇xψx,v+σvψx,v=φx,v,ψ-=Hψ+,where φ∈Xp and the unknown ψ must belong to D(TH). Let(11)λ⋆≔μ-ess infv∈V σv.For Reλ+λ⋆>0, (10) can be solved formally by (12)ψx,v=ψx-t-x,vv,ve-λ+σvt-x,v+∫0t-x,ve-λ+σvsφx-sv,vds.Moreover, if (x,v)∈Γ+, (10) becomes (13)ψ+x,v=ψ-e-λ+σvτx,v+∫0τx,ve-λ+σvsφx-sv,vds,where τ(x,v)=t+(x,v)+t-(x,v). On the other hand, for every (x,v)∈D¯×V, we have (x-t-(x,v)v,v)∈Γ- (for more details on the time numbers t+, t-, and τ, see [1]).
For the abstract formulation of (12) and (13), we define the following operators depending on the parameter λ: (14)Mλ:Lp-⟶Lp+,u⟶Mλu≔ue-λ+σvτx,v;Bλ:Lp-⟶Xp,u⟶Bλu≔ue-λ+σvt-x,v;Gλ:Xp⟶Lp+,φ⟶∫0τx,ve-λ+σvsφx-sv,vds;Cλ:Xp⟶Xp,φ⟶∫0t-x,ve-λ+σvsφx-sv,vds.Straightforward calculations using Hölder’s inequality show that all these operators are bounded on their respective spaces. More precisely, we have, for Reλ>-λ⋆,(15)Mλ≤1,Bλ≤pReλ+λ⋆-1/p,Gλ≤qReλ+λ⋆-1/q,Cλ≤1Reλ+λ⋆1p+1q=1.
1.1. Collision Operators
The collision operator K given as a perturbation of the advection transport operator TH is defined on Xp by (16)K:Xp⟶Xpψ⟶∫Vκx,v,v′ψx,v′dμv′.Note that the operator K is local in x; it describes the physics scattering and production of particles (fission), so it can be viewed as mapping: (17)K·:x∈D⟶Kx∈LLpV.We assume that K(·) is strongly measurable,(18)x∈D⟶Kxφ∈LpV is measurable for any φ∈LpV,and bounded, (19)ess supx∈DKxLLpV<∞.It follows that K defines a bounded operator on the space Lp(D×V) according to the formula(20)φ∈LpD×V(Lp(D×V)≃Lp(D;Lp(V))) and(21)KxLLpD×V≤ess supx∈DKxLLpV.The final assumption on K is (22)Kx∈KLpV almost everywhere,where K(Lp(V)) denotes the set of compact linear operators on the space Lp(V).
We give now the concept of regular collision operators introduced by Mokhtar-Kharroubi [5].
Definition 1.
A collision operator, (23)K·:x∈D⟶Kx∈LLpV,is said to be regular if K(x) is compact on Lp(V) almost everywhere on D and (24)K·:x∈D⟶LLpVis a “Bochner measurable function”.
The interest of the class of regular collision operators lies in the following lemma.
Lemma 2 (see [5, Proposition 4.1]).
A regular collision operator K can be approximated, in the uniform topology, by a sequence {Kn} of collision operators with kernels of the form (25)∑i∈Ifixgiξhiξ′,where fi∈L∞(D), gi∈Lp(V) and hi∈Lq(V) (1/p+1/q=1) (I is finite).
It is easy to observe that (1) can be written under the following abstract Cauchy problem:(26)∂ψ∂t=TH+Kψt, t>0,ψ0=ψ0.Spectral theory of transport operators has known a major development since the pioneering papers of Lehner and Wing and Jörgens in the late 1950s [6–8]. A considerable literature has been devoted to the spectral analysis of the transport operator. This one is studied by means of the nature of the parameters of the equation (nature of boundary conditions, nature of the domain of positions or velocity space, and nature of the collision operator). Let us quote, for example, [1, 4–6, 9–48].
In general, the time asymptotic behavior of solutions of (1) is analyzed under two angles: resolvent approach and the semigroup approach.
(1) Resolvent Approach. For 1<p<∞, this approach is based essentially on the compactness (or the compactness of one iterate) of the bounded linear operator (λ-TH)-1K. Indeed, Vidav [44] observed that if this condition is satisfied, it leads via an analytic Fredholm alternative to the fact that the set σ(TH+K)∩{λ∈C:Reλ>sH} (σ is the spectrum, while sH is the spectral bound of the operator TH) composed (at most) a set of isolated eigenvalues with finite algebraic multiplicities {λi}i∈J, where {λi,Reλi≥α} is a finite set for each α>sH. If p=1, it suffices to treat the weak compactness by taking into account the fact that the square of weakly compact operator on this space is compact [49, Corollary 13, p. 510]. Recall that, among relevant results in this direction, we can cite the works of Mokhtar-Kharroubi [5, 36], Latrach [24–27], and Song [43].
Thus, if TH generates a c0-semigroup (U(t); t≥0), by Dyson-Phillips theorem of perturbation, TH+K generates a c0-semigroup (V(t); t≥0) given by the following formula (see [50, Corollary 7.5, p. 29]):(27)Vtψ0=12iπlimγ→∞∫v-iγv+iγeλtλ-T-K-1ψ0dλ, t>0,where v is sufficiently large by deforming the contour of integration in Dunford’s formula. Recover the residues corresponding to the poles (eigenvalues of TH+K); we can obtain a good comprehension of the asymptotic behavior of solution when the initial data ψ0 belongs to D(TH+K)2 (unfortunately this regular condition is not natural).
(2) Semigroup Approach. Even if σ(TH+K)∩{λ∈C:Reλ>sH} is reduced to isolated eigenvalues of finite algebraic multiplicities, the set σ(V(t))∩{η∈C:|η|>esHt} can contain the continuous spectrum due to the absence of a spectral mapping theorem for the mapping λ→etλ. Vidav [45] has shown that the time asymptotic behavior of V(t)t≥0 is connected to the analysis of its spectrum and the compactness of remainder terms of the Dyson-Phillips expansion Rn(t)=∑j=n∞Uj(t) (where U0(t)=U(t) and Un(t)=∫0tU(t-s)KUn-1(s)ds for all n≥1) is an appropriate tool to exclude the eventual presence of the continuous spectrum and to restore the following spectral mapping theorem:(28)σVt∩η∈C:η>esHt=etσTH+K∩etλ:λ>sH.This technique has the advantage of not imposing any condition on the initial data; it has been used by [36, 44, 45, 51] and other authors to study the time asymptotic behavior of solutions of transport equations for absorbing boundary conditions (H=0) or ψ|Γ-=0; in other words, it has been used in the case where each neutron which arrives at a point of ∂D and coming from the interior of D disappears, and no neutron arrives from outside and where D is bounded. Many contributions have been made in this direction, showing in particular the compactness of the second-order remainder of the Dyson-Phillips expansion, sometimes through heavy calculations in the case of non absorbing boundary conditions. Recently and always for absorbing boundary conditions, dealing with regular collision operators by assuming that the domain of positions has a finite volume (not necessarily bounded), Mokhtar-Kharroubi [40] has established the compactness of the first remainder term of the Dyson-Phillips expansion on Lp(D×V) (1<p<∞). This analysis simplifies considerably the spectral analysis of transport equations and extends all known results made in the framework of the study of the compactness of the second-order remainder term; this is due to the fact that if Rn(t) is compact, thus Rn+1(t) is also compact, and it implies that (U(t))t≥0 and (V(t))t≥0 have the same essential spectra and consequently the same essential types. Unfortunately, this argument cannot be applied to the case where p=1 since its proof was obtained in the framework of L2(D×V) (and extended to Lp(D×V) space (1<p<∞) via some interpolation techniques) using some properties of Fourier transform and Hilbert-Schmidt operators. Better than that, Mokhtar-Kharroubi conjectured that the first remainder term of the Dyson-Phillips expansion R1(t), t>0 is not compact on L1(D×V); additionally, its weakly compactness is an open problem (see [5, Problem 7, p. 94]).
In this work, we study the impact of compactness results on p-independence of the asymptotic spectrum of the transport operator AH. These results are established by means of some geometrical properties of the space of positions D and the Radon measure μ having the velocity space V as a support and the natures of the collision operator K and the boundary linear operator H.