On the Fourier-Transformed Boltzmann Equation with Brownian Motion

Abstract

We establish a global existence theorem, and uniqueness and stability of solutions of the Cauchy problem for the Fourier-transformed Fokker-Planck-Boltzmann equation with singular Maxwellian kernel, which may be viewed as a kinetic model for the stochastic time-evolution of characteristic functions governed by Brownian motion and collision dynamics.

1. Introduction

In this paper, we consider the Cauchy problem for the space-homogeneous Fokker-Planck-Boltzmann equation which takes the form Here, the diffusion constant , is a nonnegative initial datum and stands for the collision term defined as for each scalar-valued function on where the collision kernel is a nonnegative function on , and denotes the area measure on the unit sphere .

In kinetic theory of a rarefied gas, the Fokker-Planck-Boltzmann equation (1) models the single-particle distribution function of its molecules which evolve under binary and elastic collision dynamics as well as Brownian motion (see below). Each pair represents the postcollision velocities of two molecules colliding with velocities .

The collision kernel is an implicitly-defined function which represents a specific type of collision dynamics in terms of the deviation angle defined by . In a physically relevant model known as the Maxwellian kernel, it is customary to assume that is supported in , bounded away from , but develops a singularity at in the form which accounts for grazing collisions in the long-range interactions.

The Maxwellian kernel is a special instance of known as the collision kernel of inverse-power potential type, and we refer to Villani’s review paper [1] for more details. Besides the physically relevant assumption (4) on , a simplified one is that referred to as Grad’s angular cutoff assumption.

The inhomogeneous Fokker-Planck-Boltzmann equation reads for the unknown density , where the space variable stands for the position. In the case when the collision kernel takes form (5) and the angular part satisfies certain cutoff assumption of type (6), let us mention some of the earlier works on the Cauchy problem for (7). In the small perturbations of the vacuum state, a global existence result is obtained by Hamdache [2]. In the context of renormalized solutions, global existence and stability of solutions with large data are established by DiPerna and Lions [3]. In the linearized context around the global Maxwellian , global existence or asymptotic behavior of solutions is investigated by Li and Matsumura [4], Xiong et al. [5], and Zhong and Li [6]. We also refer to Li [7] for the diffusive property of solutions and further references cited in the aforementioned work.

As for the homogeneous Fokker-Planck-Boltzmann equation, we are aware only of results of Goudon [8] for the global existence of a weak solution in the case when the collision kernel is given by (5) with and satisfies a singular condition of type (4). For the homogeneous Boltzmann equation, however, more extensive results are available. We refer to Arkeryd [9, 10], Goudon [8], and Villani [11] and to the references cited therein.

We recall that the Fourier transform of a complex Borel measure on is defined by which extends to any tempered distribution on via the usual functional pairing relations. If is a probability measure, that is, a nonnegative Borel measure with unit mass, is said to be a characteristic function.

From a probability theory point of view, Cauchy problem (1), with an initial probability density , could be considered as a governing equation for the time-evolution of a family of probability densities and, hence, it is natural to study the problem on the Fourier transform side for it is fundamental in probability theory to investigate a probability distribution through its characteristic function.

In [12], Bobylev discovered a remarkably simple formula for the Fourier transform of the collision term which reads for each nonzero . To simplify, we introduce the Boltzmann-Bobylev operator defined by for each complex-valued function on . In view of Bobylev’s formula, the Fourier-transformed version of (1) takes the form which is equivalent to the integral equation provided that differentiation under the integral sign was permissible.

In the theory of stochastic processes, a Markov process in any Euclidean space , with stationary independent increments, for which the characteristic functions of its continuous transition probability densities are given by the Gaussian family is known as Brownian motion or the symmetric stable Lévi process of index (see [13]). Hence, Cauchy problem (11) may be viewed as a kinetic model for the stochastic time-evolution of characteristic functions governed by Brownian motion and Maxwellian collision dynamics. For more detailed interpretations and motivations, we refer to the inspiring paper [14] of Bisi et al. which deals with Cauchy problem (11) in the inelastic setting.

In this paper, we are concerned about global existence and uniqueness and stability of solutions of Cauchy problem (11) in the space of characteristic functions. Before proceeding further, let us describe briefly some of the earlier works about the Cauchy problem for the corresponding Fourier-transformed Boltzmann equation: for which the Maxwellian kernel is assumed to satisfy the singular or noncutoff condition as described in (4).(a)It is Pulvirenti and Toscani [15] who first established a global existence of solution to (13) on the space of characteristic functions satisfying  They also proved uniqueness and stability of solutions in terms of Tanaka’s functionals related with probabilistic Wasserstein distance.(b)In [16], Toscani and Villani proved uniqueness and stability, on the same solution space, with respect to the Fourier-based metric which is a particular case of  for each where and (see also [17] for the properties of Fourier-based metrics and their applications to the Boltzmann and Fokker-Planck-Boltzmann equations in the inelastic setting).(c)In [18], Bobylev and Cercignani constructed an explicit class of self-similar solutions whose probability densities possess infinite energy for all time. Specifically, they exhibited a class of characteristic functions satisfying (13) and for all .(d)In [19], Cannone and Karch established global existence and uniqueness and stability of solutions on the space , to be explained below, which turns out to be larger than the solution space of Pulvirenti and Toscani and closely related with infinite energy solutions. In [20], Morimoto improved their work by weakening the assumptions on the kernel and providing another proof of uniqueness.

As to Cauchy problem (11), our aim is to establish global existence and uniqueness and stability of solutions on the space introduced by Cannone and Karch [19]. Following their notation, let be the set of all characteristic functions on . For , let While is not a vector space, it is a complete metric space with respect to the Fourier-based metric defined in (15) (for the proofs and further properties, see [19]). As a monotonically indexed family, the embedding holds for . Any characteristic function satisfying (14) clearly belongs to . More extensively, it can be trivially verified that if is a probability measure on such that with the additional assumption that the first-order moments vanish in the case , then . The reverse implication, however, is false as it can be seen from the Lévi characteristic function with which belongs to but

As a means of treating singularity, we follow Morimoto to consider weak integrability of the kernel in the form with . It is certainly satisfied by the true Maxwellian kernel which behaves like (4) as long as . In addition, we will consider for , which is independent of and finite under condition (20) for all . Introduced by Cannone and Karch, these quantities will serve as the stability exponents.

To state our results, we set down the precise solution spaces. Let be arbitrary. As it is customary, we denote by the space of all complex-valued functions on such that for each and the map is continuous on . By the Riemann-Lebesgue lemma, each characteristic function is continuous in and, hence, the space continuity is alluded in the definition of .

In consideration of time regularity, we will write for the space of such that , for each fixed . We put

Our main result for global existence is as follows.

Theorem 1. Assume that the collision kernel satisfies a weak integrability condition (20) for some and . Then, for any initial datum , there exists a classical solution to the Cauchy problem (11) in the space satisfying

A distinctive feature is the existence of a solution satisfying the stated maximum growth estimate which asserts in a sense that the solution stays within Brownian motion for all time.

To state our main result for stability and uniqueness, we put

Theorem 2. Under the same hypotheses on as in Theorem 1, if are solutions to Cauchy problem (11) in the space corresponding to the initial data , respectively, then, for all , In particular, for any initial datum , Cauchy problem (11) has at most one solution in the space .

Upon setting , both theorems are reduced to those of Cannone and Karch and Morimoto. In fact, due to a special structure of the Boltzmann-Bobylev operator, to be explained below, the existence theorem is an almost instant consequence of their existence theorem except for some technical points. On the other hand, the stability theorem is not so straightforward and our proof will be carried out along Gronwall-type reasonings.

As some functionals or expressions involving the space and time variables are too lengthy to put effectively, we will often abbreviate the space variables for simplicity in the sequel.

2. Preliminaries

A well-known Fourier transform formula states that and, hence, it is clear that the Gaussian family whose probability densities are self-similar Gaussian functions (see, e.g., [21]).

Lemma 3. If and , then

Proof. Observe that Since is a smooth function on with the assertion follows.

The Boltzmann-Bobylev operator defined in (10) takes the form for each characteristic function . We set in the sequel.

For a nonzero , by considering a parametrization of the unit sphere in terms of the deviation angle from , it is well known that in which and denotes the area measure on the unit circle . As it is defined in (9), the spherical variables are expressed in terms of via

The following are due to Morimoto [20, page 555]. We put which is finite under condition (20) for any .

Lemma 4. For , assume that the kernel satisfies the condition . Let and . Then, for each . Moreover,

As an application, we have the following time-continuity property.

Lemma 5. For , assume that the kernel satisfies the condition and . If and for each , then for each .

Proof. Fix a nonzero and . For any sequence with , we may write, with the aid of (31), By the estimate (34), we notice that By the continuity of , we have . Since uniformly in and the definition of gives we may apply Lebesgue’s dominated convergence theorem to evaluate the limit under the integral sign , which proves continuity at .

3. Global Existence

An important feature of the Boltzmann-Bobylev operator is that it satisfies the pointwise identity for any scalar-valued function defined on and for any scalar-valued function on , which results from for all and . As a consequence, at the formal level, it is straightforward to find that if is a solution to Cauchy problem (13) of the Fourier-transformed Boltzmann equation, then is a solution to Cauchy problem (11) of our consideration.

To be rigorous, we begin with quoting the existence theorem of Cannone and Karch [19] and Morimoto [20] in a combined manner.

Theorem 6. Assume that satisfies (20) for some . Let and . Then, there exists a unique classical solution to Cauchy problem (13) in the space .

Remark 7. In their work, Cannone and Karch constructed a unique solution on the space without mentioning time-regularity conditions. Since is not a Banach space, the meaning of a classical solution to Cauchy problem (20) is not so clear in this space. Inspecting their proof and making use of the time continuity of the Boltzmann-Bobylev operator as stated in Lemma 5, however, it is not hard to find that their solution is indeed a classical solution in the space for which the partial derivative in time is taken in the usual pointwise sense.
Let us consider an equivalent formulation of (13): where the time integration is taken in the usual Riemann sense. By Lemma 5, if and is continuous in for each fixed , then this integral is well defined for a kernel satisfying .
We will need a technical lemma in support of Theorem 6.

Lemma 8. For , let and . Assume that and is continuous in for each fixed . If is a solution to (41), then, for all ,

Proof. (i) An application of Lemma 4 yields which yields the desired estimate in view of Gronwall’s lemma.
(ii) Assuming , we apply Lemma 4 once again to find which yields the desired estimate upon combining with (i).

Proof of Theorem 1. Since the stated assumptions on and are the same as those of Theorem 6, there exists a unique solution to Cauchy problem (13) in the space . Put We will verify that is a solution to Cauchy problem (11) satisfying the stated maximum growth estimate. (i)Clearly for any fixed . Moreover, Lemma 3 gives  which implies for any fixed with  (ii)For , with an arbitrary , writing  we deduce from Lemmas 3 and 8  Thus, the map is Lipschitz continuous in for
Therefore, for the time-regularity conditions are obviously valid. In particular, Lemmas 4 and 5 imply that is well defined with for each and is continuous in for each . Clearly, . We calculate for all , where we have used (40). Thus, satisfies Cauchy problem (11). Since it is obvious that our proof of Theorem 1 is complete.

Remark 9. In our forthcoming paper [22], we study the Cauchy problem for the Boltzmann equation coupled with fractional Laplacian diffusion terms on the Fourier transform side in which we give direct proofs of global existence.

4. Uniqueness and Stability of Solutions

To proceed our proof for stability of solutions, we begin with estimating the time-growth behavior of for each solution of Cauchy problem (11) or integral equation (12).

Lemma 10. Under the same hypotheses on , , as stated in Lemma 8, if is a solution to (12), then

Proof. Writing and applying Lemma 4, it is straightforward to obtain A Gronwall-type argument yields

Proof of Theorem 2. We will prove the stated stability inequality for each with an arbitrarily fixed .
Let us consider a monotone sequence of kernels obtained from by cutting off the singularity at in the manner Since is assumed to be at least bounded away from , it is clear that each is integrable on the unit sphere, and monotonically. Setting for each , we introduce two sequences of operators , defined by Suppose that are solutions to Cauchy problem (11) with the initial data , respectively. Then, for which we denote Upon setting for and , the above identity implies Let be arbitrary. Put and For , we make use of to estimate Likewise, we make use of to estimate for all . Since , , we find for all . Henceforth, it is straightforward to deduce for all . On the other hand, Lemma 4 gives By considering similar estimate for , hence, we note for all and , where we put In view of the growth estimate of Lemma 10, and so an application of Lebesgue’s dominated convergence theorem shows as under the assumption .
Now, estimates (68) and (70) imply for all , . A standard Gronwall-type argument gives Since we notice increasingly as . Passing to the limit, we conclude for all . Letting , we finally obtain which is equivalent to the desired stability estimate on .

5. Concluding Remarks

Having established global existence and uniqueness and stability of solutions to the Fourier-transformed version of Fokker-Planck-Boltzmann equation on the space , we end our paper with a few additional remarks.(a)Concerning Theorem 1, while it asserts that there exists a solution of Cauchy problem (11) satisfying  a natural question is whether this dominating property would hold for any solution of (11). The answer is affirmative in the case when the collision kernel satisfies Grad’s angular cutoff assumption. Suppose and . If is a solution to (11) in the space , then necessarily (77) holds. As it can be proved in an elementary way, we leave its verification to the interested reader. For the singular case of , however, we were not able to draw any conclusion.(b)In the cutoff case of , it is possible to construct an explicit solution of Cauchy problem (11) by using the Wild sum method as developed in [15, 17, 23]. Assuming , if we follow the same known method, then it is straightforward to derive  where is the initial datum and  for . It can be shown plainly that if , then this explicit solution for . By uniqueness, this solution coincides with the solution of Theorem 1.(c)Concerning the asymptotic behavior of a solution to Cauchy problem (11), an important question common in kinetic theory is whether there exists a steady-state equilibrium such that as in some sense. For instance, in the inelastic case, it is shown that there exists such steady-state equilibrium for a solution of the Cauchy problem for the corresponding Fokker-Planck-Boltzmann equation (see [17] and further references therein). In the elastic case, however, it is likely that the answer would be negative in view of the pointwise behavior due to growth estimate (77). A seemingly reasonable alternative is to investigate if the solution gets close to the Gaussian in an appropriate sense.

Conflict of Interests

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

Acknowledgments

Yong-Kum Cho is supported by National Research Foundation of Korea Grant funded by the Korean Government (no. 20130301) and 2013-14 Chung-Ang University Sabbatical Research Grant.