We discuss some compactness results in spaces related to the spectral theory of neutron transport equations for general classes of collision operators and Radon measures having velocity spaces as supports covering most physical models. We show in particular that the asymptotic spectrum of the transport operator is independent of .

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 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 which models the density function of gas particles having position and velocity at time . The integral of this function gives the expectation value (statistical average) of the total mass of gas contained in the phase space . Under some assumptions, function must satisfy the Boltzmann equationcompleted 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 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 , 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 formwith initial data , where . is a smooth open subset of and is a positive Radon measure on such that and (admissible velocity space) denotes the support of . The function 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 (resp., ) with (resp., ) being the incoming (resp., outcoming) part of the phase space boundary and is a linear bounded operator from a suitable function space on to a similar one on . The classical boundary conditions (vacuum boundary, specular reflections, diffuse reflections, and periodic and mixed type boundary conditions) are special examples of our framework.

Let . We define the positive real numbers byPhysically, is the time taken by a neutron initially in with animated speed to achieve (for the first time) the boundary of .

We denote by the setwhere is the outer unit normal vector at .

Let ; we introduce the functional spaces whereThe spaces of traces are . Here is the Lebesgue measure on .

Recall that, for every , we can define the traces on ; unfortunately, these traces do not belong to . The traces lie only in or precisely in a certain weighted space (see [24], for details).

Define In this case and the associated advection operator is given as follows: with domain where the collision frequency (in other words, a positive bounded function).

Let ; consider the boundary value problemwhere and the unknown must belong to . LetFor , (10) can be solved formally by Moreover, if , (10) becomes where . On the other hand, for every , we have (for more details on the time numbers , , and , see [1]).

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 ,

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, It follows that defines a bounded operator on the space according to the formula() andThe final assumption on is where denotes the set of compact linear operators on the space .

We give now the concept of regular collision operators introduced by Mokhtar-Kharroubi [5].

Definition 1. A collision operator, 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 ]). A regular collision operator can be approximated, in the uniform topology, by a sequence of collision operators with kernels of the form where , and ( is finite).

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 [68]. 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, 46, 948].

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 , this approach is based essentially on the compactness (or the compactness of one iterate) of the bounded linear operator . Indeed, Vidav [44] observed that if this condition is satisfied, it leads via an analytic Fredholm alternative to the fact that the set ( is the spectrum, while is the spectral bound of the operator ) composed (at most) a set of isolated eigenvalues with finite algebraic multiplicities , where is a finite set for each . If , 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 , p. 510]. Recall that, among relevant results in this direction, we can cite the works of Mokhtar-Kharroubi [5, 36], Latrach [2427], and Song [43].

Thus, if generates a -semigroup , by Dyson-Phillips theorem of perturbation, generates a -semigroup given by the following formula (see [50, Corollary , p. 29]):where 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 belongs to (unfortunately this regular condition is not natural).

(2) Semigroup Approach. Even if is reduced to isolated eigenvalues of finite algebraic multiplicities, the set 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 is connected to the analysis of its spectrum and the compactness of remainder terms of the Dyson-Phillips expansion (where and for all ) 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 or ; 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 . 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 is also compact, and it implies that and have the same essential spectra and consequently the same essential types. Unfortunately, this argument cannot be applied to the case where since its proof was obtained in the framework of (and extended to space 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 is not compact on ; additionally, its weakly compactness is an open problem (see [5, Problem , 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 .

2. Main Results

2.1. Compactness Results and -Independence of

We assume that the measure satisfies the following specific geometrical property:for every and , where is the Lebesgue measure of the set and is the indicator function of .

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

We start our analysis by the following fundamental compactness result which will be used in the rest of this section.

Theorem 4. If , let 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, (i)if is bounded, then is weakly compact on ;(ii)if is a bounded convex set in and is let to be compact, then is compact on .

Proof. If , generates a -semigroup on , and then its resolvent exists as bounded linear operator satisfying where , and Thus,Obviously, if is weakly compact, then 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 . Now, under condition (29) and by applying [5, Theorem ], we obtain that is weakly compact on if is bounded. Moreover, if is convex, then, by applying [5, Theorem ], we get the compactness of the operator . 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 and if the boundary linear operator is compact, then generates a -semigroup on (see [31, Theorem ]); is the result still true for under the condition that is weakly compact multiplicative boundary operator ()?

Let , be measure spaces with -finite positive measures .

Theorem 5 (Riesz-Thorin theorem). Assume that , , for , and let be a linear operator which maps continuously into with norm . If and , , then maps continuously into with norm .

This theorem shows that the boundedness of linear operators can be interpolated between -spaces. In 1960, Krasnoselskii [52] showed that compactness can be also interpolated. Thus, we can announce the following result.

Theorem 6. Assume that , , for , and let be compact. If and , then is also compact.

By combining Theorem 6 and [53, Corollary ], the following lemma can be derived.

Lemma 7. Let , . Assume that a linear operator can be extended to bounded linear operators on such that at least one of them is power compact. Then(i) can be extended to a power compact operator on for each ;(ii)denote the extension of to by . If is power compact, then for all and the spectral projections corresponding to nonzero eigenvalues are independent of .

Let (resp., ) be the closed densely defined operator (resp., ) on , . We denote by and the bounded linear operators and defined on . Let (the asymptotic spectrum of the operator ).

Now, we establish the fundamental result of this work which describes -independence of .

Theorem 8. Under assumptions of Theorem 4, we have(i) for all ;(ii)if , we have for every positive integer and every , where designates the null space of the linear operator . As a consequence, both the geometrical multiplicity and algebraic multiplicity of are -independent.

Proof. Let ; we haveThe resolvent of can be written as the Laplace transform of as follows:Thus for such that , we getBy applying Theorem 4, we obtain the compactness of on . On the other hand, Lemma 7 implies the compactness of on and consequently for every . Hence, Gohberg-Schmul’yan theorem [22, Theorem ] shows that consists of discrete eigenvalues with finite algebraic multiplicity. Moreover, it is easy to observe that if and only if ; therefore, taking into account assertion (ii) in Lemma 7, we get that for all .
Next, following estimation (31), we obtain that . This gives that, for sufficiently large, we haveThus, for sufficiently large, it follows thatUsing the analyticity of the operator valuation function on the set for each , we get that forwe haveUsing the formula for each and , one sees thatfor .
Now, if we denote by the spectral projection corresponding to an eigenvalue of , then for sufficiently smallAccording to (40) and (41), it follows that for each Since the space of infinitely differentiable functions with compact supports is dense in , then is dense in , but these two vector spaces are finite-dimensional; henceAccording to (42) and (43), we obtain that for all and . Afterwards, since, for every , we have and , it follows that .
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) with boundedness of but based on the weak compactness of one remainder of the Dyson-Phillips expansion.
Let , where is the spectral bound of the operator in for all .

Proposition 9. Let be a regular collision operator and let such that generates a -semigroup on . If one of the remainder terms of the Dyson-Phillips series is weakly compact on , then(i) for all ;(ii)if , then for every positive integer and every . 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 such that is compact for such that . Indeed, assume that there exists such that is weakly compact on . Then, according to [5, Theorem ], is weakly compact and consequently by [5, Theorem ] the strong integralOn the other hand, we haveHence, is weakly compact on . Since the Laplace transform of is nothing but , this gives the weak compactness of for such that , where is the type of ; by analytic arguments, we obtain that is weakly compact for such that and implies compactness of which gives the needed result.

Now, we focus our study on the case of slab geometry.

Theorem 10. If (the Lebesgue measure on ) and is a bounded boundary linear operator from to , then(i) for all ;(ii)if , then for every positive integer and every . As a consequence, both the geometrical multiplicity and algebraic multiplicity of are -independent.

Proof. Here, the time of sojourn of particles in is bounded from below by ; indeed, in this case we have . As an immediate consequence, generates a -semigroup for any boundary linear operator [29, Remark ]. Moreover, we havewhere is a positive constant depending on and withOn the other hand, we have is compact for all [25, Theorem ]. By adopting the same techniques given in the proof of Theorem 8, we obtain the desired result.

Remark 11. We note that is not weakly compact in general on (see [18]).

Question 2. In the case of slab geometry, is weakly compact on under the assumption that is a diffuse (nonatomic) measure on ?

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.