Some Compactness and Interpolation Results for Linear Boltzmann Equation

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 ∈ R at time t ∈ R. The integral of this function


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  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 (, V, ) which models the density function of gas particles having position  ∈  and velocity V ∈ R 3 at time  ∈ R. The integral of this function ∫ ∫ ×R 3 (, V, ) V gives the expectation value (statistical average) of the total mass of gas contained in the phase space ×R 3 .Under some assumptions, function  must satisfy the Boltzmann equation   (, V, ) = −V ⋅ ∇   (, V, ) +  ( (, ⋅, )) (V) (⋆) completed 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 ((, ⋅, )), which is bilinear, describes the mechanism of collisions.A solution to the initial boundary value problem for (⋆) and a proof of -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 warmup 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.
with initial data (, V, 0) =  0 (, V), where (, V) ∈  × . is a smooth open subset of R  and (⋅) is a positive Radon measure on R  such that ({0}) = 0 and  (admissible velocity space) denotes the support of .The function (; V; ) describes the distribution of the neutrons in a nuclear reactor occupying the region .The functions (⋅) and (⋅, ⋅, ⋅) 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  satisfying where  − (resp.,  + ) is the restriction of  to Γ − (resp., Γ + ) with Γ − (resp., Γ + ) being the incoming (resp., outcoming) part of the phase space boundary and  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 (, V) ∈  × .We define the positive real numbers  ± (; V) by  ± (, V) = sup { > 0;  ± V ∈ , ∀0 <  < } . ( Physically,  ± (, V) is the time taken by a neutron initially in  ∈  with animated speed ±V to achieve (for the first time) the boundary of .
Recall that, for every  ∈   , we can define the traces  ± on Γ ± ; unfortunately, these traces do not belong to  ±  .The traces lie only in  ± ,loc or precisely in a certain weighted   space (see [2][3][4], for details). Define In this case  ∈ L( +  ,  −  ) (1 ≤  < ∞) and the associated advection operator   is given as follows: with domain where the collision frequency (⋅) ∈  ∞ + () (in other words, a positive bounded function).
For the abstract formulation of ( 12) and ( 13), we define the following operators depending on the parameter : Straightforward calculations using Hölder's inequality show that all these operators are bounded on their respective spaces.More precisely, we have, for Re > − ⋆ , 1.1.Collision Operators.The collision operator  given as a perturbation of the advection transport operator   is defined on   by Note that the operator  is local in ; it describes the physics scattering and production of particles (fission), so it can be viewed as mapping: We assume that (⋅) is strongly measurable, and bounded, ess sup It follows that  defines a bounded operator on the space   ( × ) according to the formula (  ( × ) ≃   (;   ())) and The final assumption on  is where K(  ()) denotes the set of compact linear operators on the space   ().
We give now the concept of regular collision operators introduced by Mokhtar-Kharroubi [5].
is said to be regular if () is compact on   () almost everywhere on  and is 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 {𝐾 𝑛 } of collision operators with kernels of the form
where It is easy to observe that (1) can be written under the following abstract Cauchy problem: 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][7][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][5][6].
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 <  < ∞, this approach is based essentially on the compactness (or the compactness of one iterate) of the bounded linear operator ( −   ) −1 .Indeed, Vidav [44] observed that if this condition is satisfied, it leads via an analytic Fredholm alternative to the fact that the set (  + ) ∩ { ∈ C : Re >   } ( is the spectrum, while   is the spectral bound of the operator   ) composed (at most) a set of isolated eigenvalues with finite algebraic multiplicities {  } ∈ , where {  , Re  ≥ } is a finite set for each  >   .If  = 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][25][26][27], and Song [43].
Thus, if   generates a  0 -semigroup (();  ≥ 0), by Dyson-Phillips theorem of perturbation,   +  generates a  0 -semigroup (();  ≥ 0) given by the following formula (see [50, Corollary 7.5, p. 29]): where V is sufficiently large by deforming the contour of integration in Dunford's formula.Recover the residues corresponding to the poles (eigenvalues of   + ); we can obtain a good comprehension of the asymptotic behavior of solution when the initial data  0 belongs to (  +) 2 (unfortunately this regular condition is not natural).
(2) Semigroup Approach.Even if (  + ) ∩ { ∈ C : Re >   } is reduced to isolated eigenvalues of finite algebraic multiplicities, the set (()) ∩ { ∈ C : || >     } can contain the continuous spectrum due to the absence of a spectral mapping theorem for the mapping  →   .Vidav [45] has shown that the time asymptotic behavior of () ≥0 is connected to the analysis of its spectrum and the compactness of remainder terms of the Dyson-Phillips expansion   () = ∑ ∞ =   () (where  0 () = () and   () = ∫  0 ( − ) −1 () for all  ≥ 1) is an appropriate tool to exclude the eventual presence of the continuous spectrum and to restore the following spectral mapping theorem: 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 ( = 0) or  |Γ − = 0; in other words, it has been used in the case where each neutron which arrives at a point of  and coming from the interior of  disappears, and no neutron arrives from outside and where  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   (×) (1 <  < ∞).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   () is compact, thus  +1 () is also compact, and it implies that (()) ≥0 and (()) ≥0 have the same essential spectra and consequently the same essential types.Unfortunately, this argument cannot be applied to the case where  = 1 since its proof was obtained in the framework of  2 ( × ) (and extended to   ( × ) space (1 <  < ∞) 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  1 (),  > 0 is not compact on  1 ( × ); 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 -independence of the asymptotic spectrum of the transport operator   .These results are established by means of some geometrical properties of the space of positions  and the Radon measure  having the velocity space  as a support and the natures of the collision operator  and the boundary linear operator .

Compactness Results
and -Independence of   (  ).We assume that the measure  satisfies the following specific geometrical property: for every 0 <  1 <  2 < ∞ and  3 < ∞, where || is the Lebesgue measure of the set  and   is the indicator function of .We start our analysis by the following fundamental compactness result which will be used in the rest of this section.

Theorem 4. If
, ‖‖ < 1 be a weakly compact boundary linear operator and let  be a Radon measure satisfying the condition (29).Assume that the collision operator  is regular.Thus, Proof.If ‖‖ < 1,   generates a  0 -semigroup (  ();  ≥ 0) on  1 , and then its resolvent exists as bounded linear operator satisfying where Γ   = ∑ ≥0   (  )    , and Thus, Obviously, if  is weakly compact, then Γ   is weakly compact.On the other hand, it is easy to observe that   is nothing but the resolvent of the streaming operator with vacuum boundary conditions  0 .Now, under condition (29) and by applying [5, Theorem 4.4()], we obtain that ( −  0 ) −1  is weakly compact on  1 if  is bounded.Moreover, if  is convex, then, by applying [5, Theorem 4.4()], we get the compactness of the operator ( −  0 ) −1 .Since the compactness and weak compactness property concerning bounded linear operators is stable under summation, we obtain the desired result.
Question 1.It is known that for 1 <  < ∞ and if the boundary linear operator  is compact, then   generates a  0 -semigroup (  ();  ≥ 0) on   (see [31,Theorem 6.8]); is the result still true for  = 1 under the condition that  is weakly compact multiplicative boundary operator (‖‖ ≥ 1)?
By combining Theorem 6 and [53, Corollary 1.6.2], the following lemma can be derived.
Assume that a linear operator  :   0 () ∩   1 () →   0 () ∩   1 () can be extended to bounded linear operators on    (), ( = 0, 1) such that at least one of them is power compact.Then (i)  can be extended to a power compact operator on   () for each  ∈ ( 0 ,  1 ); (ii) denote the extension of  to   () by   .If    is power compact, then (  ) = (   ) for all  ∈ ( 0 ,  1 ) and the spectral projections corresponding to nonzero eigenvalues are independent of . ).Now, we establish the fundamental result of this work which describes -independence of    (  ).
Theorem 8.Under assumptions of Theorem 4, we have ) for every positive integer  and every  > 1, where N() designates the null space of the linear operator .As a consequence, both the geometrical multiplicity and algebraic multiplicity of  are -independent.

Journal of Function Spaces
Using the analyticity of the operator valuation function  → ( − ( −    ) −1   ) −1 on the set { ∈ C/Re > − ⋆ } \    (  ) for each  ≥ 1, we get that for we have Using the formula for  ∈ { ∈ C/Re > − ⋆ } \    (  ).Now, if we denote by P  (   ) the spectral projection corresponding to an eigenvalue  of    , then for  > 0 sufficiently small According to (40) and (41), it follows that for each  ∈    (  ) =  1  (  ) Since the space of infinitely differentiable functions with compact supports C ∞ 0 ( × ) is dense in   , then ), but these two vector spaces are finite-dimensional; hence According to (42) In the spirit of the above theorem, we can prove the following result without weak compactness hypothesis on the boundary linear operator  and the geometrical property (29) ) for every positive integer  and every  > 1.As a consequence, both the geometrical multiplicity and algebraic multiplicity of  are independent.
Proof.Following the proof of Theorem 8, it suffices to show that there exists  ≥ 1 such that (( −   ) −1 )  is compact for  ∈ C such that Re > (  ).Indeed, assume that there exists  ≥ 1 such that    () is weakly compact on  1 .Then, according to [5, Theorem where  is the type of (  ();  ≥ 0); by analytic arguments, we obtain that (( −   ) −1 )  ( −   ) −1 is weakly compact for  ∈ C such that Re > (  ) and implies compactness of (( −   ) −1 ) 2+2 which gives the needed result.Now, we focus our study on the case of slab geometry.

Remark 3 .
As indicated in[5, Remark 4.3], the above condition is satisfied by the Lebesgue measure on R  or on spheres (multigroup model).