# On the Fourier-Transformed Boltzmann Equation with Brownian Motion

Authors: Yong-Kum Cho; Eunsil Kim
Journal: Abstract and Applied Analysis
 (2015)

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## 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.


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## Body

## 1. Introduction

In this paper, we consider the Cauchy problem for the space-homogeneous Fokker-Planck-Boltzmann equation which takes the form(1)∂tfv,t=Qf,fv,t+νΔfv,tforv,t∈R3×0,∞,fv,0=f0vforv∈R3.Here, the diffusion constant ν≥0, f0 is a nonnegative initial datum and Q(f) stands for the collision term defined as(2)Qf,fv=∫R3‍∫S2‍bk·σfv′fv∗′-fvfv∗dσdv∗for each scalar-valued function f on R3 where(3)v′=v+v∗2+v-v∗2σ,v∗′=v+v∗2-v-v∗2σ,k=v-v∗v-v∗,the collision kernel b is a nonnegative function on [-1,1], and dσ denotes the area measure on the unit sphere S2.In kinetic theory of a rarefied gas, the Fokker-Planck-Boltzmann equation (1) models the single-particle distribution function f of its molecules which evolve under binary and elastic collision dynamics as well as Brownian motion (see below). Each pair (v′,v∗′) represents the postcollision velocities of two molecules colliding with velocities (v,v∗).The collision kernelb is an implicitly-defined function which represents a specific type of collision dynamics in terms of the deviation angle θ defined by cos⁡θ=k·σ. In a physically relevant model known as the Maxwellian kernel, it is customary to assume that b(cos⁡θ) is supported in [0,π/2], bounded away from θ=0, but develops a singularity at θ=0 in the form(4)bcos⁡θsinθ~θ-3/2asθ⟶0+,which accounts for grazing collisions in the long-range interactions.The Maxwellian kernel is a special instance of(5)B=v-v∗λbcos⁡θ-3<λ≤2,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 b, a simplified one is that(6)bL1S2=2π∫0π/2‍bcos⁡θsinθdθ<+∞,referred to as Grad’s angular cutoff assumption.The inhomogeneous Fokker-Planck-Boltzmann equation reads(7)∂tf+v·∇xf=Qf,f+νΔvfinR3×R3×(0,∞)for the unknown density f=f(x,v,t), where the space variable x∈R3 stands for the position. In the case when the collision kernel B takes form (5) and the angular part b 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 M(v)=(2π)-3/2exp⁡(-|v|2/2), 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 -3<λ<-2 and b 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 measureμ on R3 is defined by (8)μ^ξ=∫R3‍e-iξ·vdμvξ∈R3,which extends to any tempered distribution on R3 via the usual functional pairing relations. If μ is a probability measure, that is, a nonnegative Borel measure with unit mass, μ^ is said to be a characteristic function.From a probability theory point of view, Cauchy problem (1), with an initial probability density f0, could be considered as a governing equation for the time-evolution of a family of probability densities {f(·,t)}t≥0 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(9)Qf,f^ξ=∫S2‍bξ·σξf^ξ+f^ξ--f^ξf^0dσ,ξ+=ξ+ξσ2,ξ-=ξ-ξσ2for each nonzero ξ∈R3. To simplify, we introduce the Boltzmann-Bobylev operator B defined by(10)Bϕξ=∫S2‍bξ·σξϕξ+ϕξ--ϕξϕ0dσfor each complex-valued function ϕ on R3. In view of Bobylev’s formula, the Fourier-transformed version of (1) takes the form(11)∂t+νξ2ϕξ,t=Bϕξ,tforξ,t∈R3×0,∞,ϕξ,0=ϕ0ξforξ∈R3,which is equivalent to the integral equation(12)ϕξ,t=e-νξ2tϕ0ξ+∫0t‍e-νξ2t-τBϕξ,τdτ,provided that differentiation under the integral sign was permissible.In the theory of stochastic processes, a Markov processXtt≥0 in any Euclidean space Rn, with stationary independent increments, for which the characteristic functions of its continuous transition probability densities are given by the Gaussian family {e-|ξ|2t}t≥0 is known as Brownian motion or the symmetric stable Lévi process of index 2 (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:(13)∂tϕξ,t=Bϕξ,tforξ,t∈R3×0,∞,ϕξ,0=ϕ0ξforξ∈R3for which the Maxwellian kernel b 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 ϕ satisfying(14)ϕ0=1,∇ϕ0=0,Δϕ0=-3.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 d2 which is a particular case of(15)dαf,g=ϕ-ψα=sup⁡ξ∈R3ϕξ-ψξξαfor eachα≥0 where ϕ=f^ and ψ=g^ (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 Φ(ξ,t) satisfying (13) and ΔΦ(0,t)=-∞ for all t≥0.(d)
In [19], Cannone and Karch established global existence and uniqueness and stability of solutions on the space Kα, 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 K be the set of all characteristic functions on R3. For 0<α≤2, let(16)Kα=ϕ∈K:ϕ-1α=sup⁡ξ∈R3ϕξ-1ξα<+∞.While Kα is not a vector space, it is a complete metric space with respect to the Fourier-based metric dα defined in (15) (for the proofs and further properties, see [19]). As a monotonically indexed family, the embedding(17){1}⊂Kβ⊂Kα⊂Kholds for 0<α≤β≤2. Any characteristic function ϕ satisfying (14) clearly belongs to K2. More extensively, it can be trivially verified that if μ is a probability measure on R3 such that(18)∫R3‍vαμdv<+∞with the additional assumption that the first-order moments vanish in the case 1<α≤2, then μ^∈Kα. The reverse implication, however, is false as it can be seen from the Lévi characteristic function μ^(ξ)=e-|ξ|α with 0<α<2 which belongs to Kα but(19)∫R3‍vαμdv=+∞.As a means of treating singularity, we follow Morimoto to consider weak integrability of the kernelb in the form(20)∫0π/2‍bcos⁡θsinθsinα0θ2dθ<+∞with 0<α0≤2. It is certainly satisfied by the true Maxwellian kernel b which behaves like (4) as long as α0>1/2. In addition, we will consider(21)λα=∫S2‍bξ·σξξ+α+ξ-αξα-1dσ=2π∫0π/2‍bcos⁡θsinθ×cos⁡αθ2+sinαθ2-1dθfor 0<α≤2, which is independent of ξ≠0 and finite under condition (20) for all α0≤α≤2. Introduced by Cannone and Karch, these quantities will serve as the stability exponents.To state our results, we set down the precise solution spaces. LetT>0 be arbitrary. As it is customary, we denote by C([0,T];Kα) the space of all complex-valued functions ϕ on R3×[0,T] such that ϕ(·,t)∈Kα(R3) for each t∈[0,T] and the map t↦∥ϕ(t)-1∥α is continuous on [0,T]. By the Riemann-Lebesgue lemma, each characteristic function is continuous in R3 and, hence, the space continuity is alluded in the definition of C([0,T];Kα).In consideration of time regularity, we will writeΩα(R3×[0,T]) for the space of ϕ∈C([0,T];Kα) such that ϕ(ξ,·)∈C([0,T]), ∂tϕ(ξ,·)∈C((0,T)) for each fixed ξ∈R3. We put (22)ΩαR3×0,∞=⋃T>0‍ΩαR3×0,T.Our main result for global existence is as follows.Theorem 1.
Assume that the collision kernelb satisfies a weak integrability condition (20) for some 0<α0≤2 and α0≤α≤2. Then, for any initial datum ϕ0∈Kα, there exists a classical solution ϕ to the Cauchy problem (11) in the space Ωα(R3×[0,∞)) satisfying (23)ϕξ,t≤e-νξ2t∀ξ,t∈R3×0,∞.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(24)ΩναR3×0,∞=e-νξ2tϕ∈ΩαR3×0,∞:ϕξ,t≤e-νξ2t∀ξ,t∈R3×0,∞e-νξ2t.Theorem 2.
Under the same hypotheses onα,b as in Theorem 1, if ϕ,ψ are solutions to Cauchy problem (11) in the space Ωνα(R3×[0,∞)) corresponding to the initial data ϕ0,ψ0∈Kα, respectively, then, for all t≥0,(25)sup⁡ξ∈R3eνξ2tϕξ,t-ψξ,tξα≤eλαtsup⁡ξ∈R3ϕ0ξ-ψ0ξξα.In particular, for any initial datum ϕ0∈Kα, Cauchy problem (11) has at most one solution in the space Ωνα(R3×[0,∞)).Upon settingν=0, 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(26)e-ξ2t=14πt3/2∫R3‍e-iξ·ve-v2/4tdvξ∈R3,t>0and, hence, it is clear that the Gaussian family {e-|ξ|2t}t≥0⊂K whose probability densities are self-similar Gaussian functions (see, e.g., [21]).Lemma 3.
If0<α≤2 and t>0, then (27)sup⁡ξ∈R31-e-ξ2tξα≤tα/2.Proof.
Observe that(28)sup⁡ξ∈R31-e-ξ2tξα=tα/2·sup⁡r>0gαrwithgαr=1-e-rrα/2.Since gα is a smooth function on (0,∞) with (29)gαr≤min⁡r1-α/2,1,limr→0+gαr=limr→∞gαr=0,the assertion follows.The Boltzmann-Bobylev operator defined in (10) takes the form(30)Bϕξ=∫S2‍bξ·σξϕξ+ϕξ--ϕξdσfor each characteristic function ϕ. We set B(ϕ)(0)=0 in the sequel.For a nonzeroξ∈R3, by considering a parametrization of the unit sphere in terms of the deviation angle from ξ/|ξ|, it is well known that(31)Bϕξ=∫0π/2‍bcos⁡θsinθ×∫S1(ξ)‍ϕξ+ϕξ--ϕξdωdθin which S1(ξ)=S2∩ξ⊥ and dω denotes the area measure on the unit circle S1⊂R3. As it is defined in (9), the spherical variables ξ+,ξ- are expressed in terms of θ,ω via (32)σ=cos⁡θξξ+sinθω.The following are due to Morimoto [20, page 555]. We put(33)μα=2π∫0π/2‍bcos⁡θsinθsinαθ2dθ,which is finite under condition (20) for any α0≤α≤2.Lemma 4.
For0<α≤2, assume that the kernel b satisfies the condition μα<+∞. Let ϕ∈Kα and ξ∈R3-{0}. Then,(34)∫S1ξ‍ϕξ+ϕξ--ϕξdω≤16πϕ-1αξαsinαθ2for each θ∈(0,π/2]. Moreover,(35)Bϕξ≤8μαϕ-1αξα.As an application, we have the following time-continuity property.Lemma 5.
For0<α≤2, assume that the kernel b satisfies the condition μα<+∞ and T>0. If ϕ∈C([0,T];Kα) and ϕ(ξ,·)∈C([0,T]) for each ξ∈R3, then B(ϕ)(ξ,·)∈C([0,T]) for each ξ∈R3.Proof.
Fix a nonzeroξ∈R3 and t0∈[0,T]. For any sequence (tn)⊂[0,T] with tn→t0, we may write, with the aid of (31),(36)Bϕξ,tn=∫S2‍bξ·σξϕξ+,tnϕξ-,tn-ϕξ,tndσ=∫0π/2‍bcos⁡θsinθAnξ,θdθwhereAnξ,θ=∫S1ξ‍ϕξ+,tnϕξ-,tn-ϕξ,tndω.By the estimate (34), we notice that(37)Anξ,θ≤16πϕtn-1αξαsinαθ2Anξ,θ≤16πCαTξαsinαθ2whereCαT=max⁡t∈0,Tϕ(t)-1α.By the continuity of t↦∥ϕ(t)-1∥α, we have Cα(T)<+∞. Since (38)bcos⁡θsinθAnξ,θ≤16πCα(T)ξαbcos⁡θsinθsinαθ2≡Aξ,θuniformly in n and the definition of μα gives (39)∫0π/2‍Aξ,θdθ=8μαCαTξα<+∞,we may apply Lebesgue’s dominated convergence theorem to evaluate the limit under the integral sign limn→∞B(ϕ)(ξ,tn)=B(ϕ)(ξ,t0), which proves continuity at t0.


## 3. Global Existence

An important feature of the Boltzmann-Bobylev operatorB is that it satisfies the pointwise identity(40)eh(t)ξ2Bϕξ,t=Behtξ2ϕξ,tfor any scalar-valued function h defined on [0,∞) and for any scalar-valued function ϕ on R3×[0,∞), which results from |ξ+|2+|ξ-|2=|ξ|2 for all ξ∈R3 and σ∈S2. As a consequence, at the formal level, it is straightforward to find that if ϕ is a solution to Cauchy problem (13) of the Fourier-transformed Boltzmann equation, then e-ν|ξ|2tϕ 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 thatb satisfies (20) for some 0<α0≤2. Let α0≤α≤2 and ϕ0∈Kα. Then, there exists a unique classical solution ϕ to Cauchy problem (13) in the space Ωα(R3×[0,∞)).Remark 7.
In their work, Cannone and Karch constructed a unique solution on the spaceC[0,∞);Kα without mentioning time-regularity conditions. Since Kα 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 ΩαR3×[0,∞) for which the partial derivative in time is taken in the usual pointwise sense.
Let us consider an equivalent formulation of (13):(41)ϕξ,t=ϕ0ξ+∫0t‍Bϕξ,τdτ,where the time integration is taken in the usual Riemann sense. By Lemma 5, if ϕ∈C[0,T];Kα and ϕ(ξ,t) is continuous in t for each fixed ξ, then this integral is well defined for a kernel b satisfying μα<+∞.
We will need a technical lemma in support of Theorem6.Lemma 8.
For0<α≤2, let μα<+∞ and ϕ0∈Kα. Assume that ϕ∈C[0,T];Kα and ϕ(ξ,t) is continuous in t∈[0,T] for each fixed ξ. If ϕ is a solution to (41), then, for all s,t∈[0,T],(42)iϕt-1α≤e8μαtϕ0-1α,iiϕt-ϕsα≤8μαe8μαTϕ0-1αt-s.Proof.
(i) An application of Lemma4 yields(43)ϕt-1α≤ϕ0-1α+8μα∫0t‍ϕτ-1αdτ,which yields the desired estimate in view of Gronwall’s lemma.
(ii) Assumings<t, we apply Lemma 4 once again to find (44)ϕt-ϕsα≤∫st‍sup⁡ξ∈R3Bϕξ,τξαdτ≤8μαmax⁡τ∈0,Tϕτ-1αt-s,which yields the desired estimate upon combining with (i).Proof of Theorem1.
Since the stated assumptions onb and ϕ0 are the same as those of Theorem 6, there exists a unique solution ϕ to Cauchy problem (13) in the space ΩαR3×[0,∞). Put (45)Φξ,t=e-νξ2tϕξ,t.We will verify that Φ is a solution to Cauchy problem (11) satisfying the stated maximum growth estimate.(i)
ClearlyΦ(·,t)∈K for any fixed t≥0. Moreover, Lemma 3 gives(46)Φξ,t-1ξα≤e-νξ2tϕξ,t-1ξα+1-e-νξ2tξα≤ϕt-1α+νtα/2,which impliesΦ(·,t)∈Kα for any fixed t≥0 with (47)Φt-1α≤e8μαtϕ0-1α+νtα/2.(ii)
Fors,t∈[0,T], with an arbitrary T>0, writing (48)Φt-Φs=-e-νξ2s1-e-νξ2t-sϕt+e-νξ2sϕt-ϕs,we deduce from Lemmas3 and 8(49)Φt-Φsα≤νt-sα/2+ϕt-ϕsα≤CTt-sα/2whereCT=να/2+8μαe8μαTϕ0-1αT1-α/2.Thus, the mapt↦Φ(t)-1α is Lipschitz continuous in [0,T] for (50)Φt-1α-Φs-1α≤Φt-Φsα.
Therefore,Φ∈Ωα(R3×[0,∞)) for the time-regularity conditions are obviously valid. In particular, Lemmas 4 and 5 imply that B(Φ) is well defined with (51)BΦξ,t≤8μαΦt-1αξαfor each (ξ,t)∈R3×[0,∞) and B(Φ)(ξ,t) is continuous in t for each ξ. Clearly, Φ(ξ,0)=ϕ0(ξ). We calculate (52)∂tΦξ,t=-νξ2e-νξ2tϕξ,t+e-νξ2t∂tϕξ,t=-νξ2Φξ,t+e-νξ2tBϕξ,t=-νξ2Φξ,t+BΦξ,tfor all (ξ,t)∈R3×(0,∞), where we have used (40). Thus, Φ satisfies Cauchy problem (11). Since it is obvious that (53)Φξ,t≤e-νξ2t∀ξ,t∈R3×0,∞,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∥ϕ(t)-1∥α for each solution ϕ of Cauchy problem (11) or integral equation (12).Lemma 10.
Under the same hypotheses onα, b, ϕ as stated in Lemma 8, if ϕ is a solution to (12), then (54)ϕt-1α≤e8μαtϕ0-1α+νtα/2t≥0.Proof.
Writing(55)ϕξ,t-1ξα=e-νξ2tϕ0ξ-1ξα+∫0t‍e-νξ2t-τBϕξ,τξαdτand applying Lemma 4, it is straightforward to obtain(56)ϕt-1α≤ϕ0-1α+νtα/2+8μα∫0t‍ϕτ-1αdτ.A Gronwall-type argument yields (57)ϕt-1α≤e8μαtϕ0-1α+ανα/22∫0t‍e-8ματτα/2-1dτ≤e8μαtϕ0-1α+νtα/2.Proof of Theorem2.
We will prove the stated stability inequality for eacht∈[0,T] with an arbitrarily fixed T>0.
Let us consider a monotone sequence(bn) of kernels obtained from b by cutting off the singularity at θ=0 in the manner(58)bncos⁡θ=bcos⁡θχ1/n,π/2θ,n=1,2,….Since b is assumed to be at least bounded away from θ=0, it is clear that each bn is integrable on the unit sphere, bn≤b and bn→b monotonically. Setting bnr=b-bn for each n, we introduce two sequences of operators (Gn), (Rn) defined by (59)Gnϕξ=∫S2‍bnξ·σξϕξ+ϕξ-dσ,Rnϕξ=∫S2‍bnrξ·σξϕξ+ϕξ--ϕξdσ.
Suppose thatϕ,ψ∈Ωνα(R3×[0,∞)) are solutions to Cauchy problem (11) with the initial data ϕ0,ψ0∈Kα, respectively. Then, (60)∂tϕ-ψ+bn1+νξ2ϕ-ψ=Gnϕ-Gnψ+Rnϕ-Rnψfor which we denote (61)bn1=2π∫1/nπ/2‍bcos⁡θsinθdθ<+∞.Upon setting (62)Uξ,t=eνξ2tϕξ,t-ψξ,tξαfor ξ≠0 and U(0,t)=0, the above identity implies(63)∂tebn1tUξ,t≤ebn1+νξ2t×Gnϕ-Gnψξα+Rnϕ-Rnψξα.
Letρ>0 be arbitrary. Put Uρ(t)=sup⁡|ξ|≤ρ|U(ξ,t)| and(64)γn,α=∫S2‍bnξ·σξξ+α+ξ-αξαdσ=2π∫1/nπ/2‍bcos⁡θsinθcos⁡αθ2+sinαθ2dθ.For |ξ|≤ρ, we make use of |ξ+|≤|ξ| to estimate (65)ϕξ+,t-ψξ+,t≤e-νξ+2tξ+α×eνξ+2tϕξ+,t-ψξ+,tξ+α≤e-νξ+2tξ+αUρt.Likewise, we make use of |ξ-|≤|ξ| to estimate (66)ϕξ-,t-ψξ-,t≤e-νξ-2tξ-αUρtfor all |ξ|≤ρ. Since |ϕ(ξ,t)|≤e-ν|ξ|2t, |ψ(ξ,t)|≤e-ν|ξ|2t, we find (67)ϕξ+,tϕξ-,t-ψξ+,tψξ-,t≤e-νξ2tξ+α+ξ-αUρtfor all |ξ|≤ρ. Henceforth, it is straightforward to deduce(68)eνξ2tGnϕ-Gnψξα≤γn,αUρtfor all |ξ|≤ρ. On the other hand, Lemma 4 gives (69)Rnϕξ,t=16πϕt-1αξα×∫01/n‍bcos⁡θsinθsinαθ2dθ.By considering similar estimate for Rn(ψ), hence, we note(70)eνξ2tRnϕ-Rnψξα≤Mnfor all |ξ|≤ρ and t∈[0,T], where we put (71)Mn=Mρ,T∫01/n‍bcos⁡θsinθsinαθ2dθ,Mρ,T=16πeνρ2T×max⁡t∈0,Tϕt-1α+ψt-1α.In view of the growth estimate of Lemma 10, (72)Mρ,T≤16πeνρ2+8μαT×ϕ0-1α+ψ0-1α+2νTα/2<+∞,and so an application of Lebesgue’s dominated convergence theorem shows Mn→0 as n→∞ under the assumption μα<+∞.
Now, estimates (68) and (70) imply(73)∂tebn1tUξ,t≤γn,αebn1tUρt+Mnebn1tfor all |ξ|≤ρ, t∈[0,T]. A standard Gronwall-type argument gives (74)Uρt≤eγn,α-bn1tUρ0+Mnγn,α-bn1eγn,α-bn1t-1.Since (75)γn,α-bn1=2π∫1/nπ/2‍bcos⁡θsinθ×cos⁡αθ2+sinαθ2-1dθ,we notice 0<γn,α-∥bn∥1→λα increasingly as n→∞. Passing to the limit, we conclude Uρ(t)≤eλαtUρ(0) for all t∈[0,T]. Letting ρ→+∞, we finally obtain (76)sup⁡ξ∈R3Uξ,t≤eλαtsup⁡ξ∈R3Uξ,0,which is equivalent to the desired stability estimate on [0,T].


## 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 spaceKα, we end our paper with a few additional remarks.(a)
Concerning Theorem1, while it asserts that there exists a solution ϕ of Cauchy problem (11) satisfying(77)|ϕ(ξ,t)|≤e-ν|ξ|2t∀ξ,t∈R3×0,∞,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 b satisfies Grad’s angular cutoff assumption.Supposeb∈L1(S2) and ϕ0∈K. If ϕis a solution to (11) in the space C([0,∞);K), 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 ofb, however, we were not able to draw any conclusion.(b)
In the cutoff case ofb, 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 ∥b∥L1(S2)=1, if we follow the same known method, then it is straightforward to derive(78)ϕξ,t=e-1+νξ2t∑n=0∞‍unξ1-e-tn,whereu0=ϕ0 is the initial datum and(79)un+1ξ=1n+1∑j=0n‍∫S2‍bξ·σξujξ+un-jξ-dσforn=0,1,…. It can be shown plainly that if ϕ0∈Kα, then this explicit solution ϕ∈Kα for 0<α≤2. By uniqueness, this solution coincides with the solution of Theorem 1.(c)
Concerning the asymptotic behavior of a solutionϕ to Cauchy problem (11), an important question common in kinetic theory is whether there exists a steady-state equilibrium ϕ∞ such that ϕ→ϕ∞ as t→∞ 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 ϕ(ξ,t)→0 due to growth estimate (77). A seemingly reasonable alternative is to investigate if the solution gets close to the Gaussian e-ν|ξ|2t in an appropriate sense.



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