Global Existence and Decay of Solutions to the Fokker-Planck-Boltzmann Equation

The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.


Introduction and Main Results
The Fokker-Planck-Boltzmann equation models the motion of particles in a thermal bath where the bilinear interaction is one of the main characters [2,3,26]. Mathematically, the Fokker-Planck-Boltzmann equation takes the following form: ∂ t f + ξ · ∇ x f = Q(f, f ) + ǫ∇ ξ · (ξf ) + κ∆ ξ f, (1.1) where the nonnegative unknown function f = f (t, x, ξ) represents the density of particles at position x ∈ R 3 and time t ≥ 0 with velocity ξ ∈ R 3 and ǫ, κ are given nonnegative constants. The collision operator Q is a bilinear operator which acts only on the velocity variables ξ and is local in (t, x) as Here ξ, ξ * and ξ ′ , ξ ′ * are the velocities of a pair of particles before and after collision. we assume these collisions to be elastic so that · ω]ω, ω ∈ S 2 . The Boltzmann collision kernel q(ξ − ξ * , ω) for a monatomic gas is, on physical grounds, a non-negative function which only depends on the relative velocity |ξ−ξ * | and on the angle θ through cos θ = ω·(ξ−ξ * )/|ξ−ξ * |.
We consider the Cauchy problem of (1.1) with prescribed initial data Throughout this manuscript, we assume that ǫ = κ > 0 such that the global Maxwellian M = (2π) −3/2 e −|ξ| 2 /2 is an equilibrium state of (1.1) and the collision kernels satisfy Grad's angular cut-off assumption: Our goal in this paper is to obtain the global existence and optimal temporal decay estimates of classical solutions for (1. It is well known that for the linearized collision operator L, one has where the collision frequency is and the operator K is defined by |ξ − ξ * | γ q 0 (θ)M 1/2 (ξ * )M 1/2 (ξ)u(ξ * )dωdξ * .
Furthermore, the operator L is non-positive, the null space of L is the five dimensional space and −L is locally coercive in the sense that there is a positive constant λ 0 such that (see [4], [17], [27]) holds for u = u(ξ), where I means the identity operator and P denotes its ξ-projection from L 2 ξ (R 3 ) onto the null space N . As in [18], for any function u(t, x, ξ), we can write P as (1.8) Here, Pu and {I − P}u is called the macroscopic component and the microscopic component of u(t, x, ξ), respectively. For later use, one can rewrite P as Notations. Throughout this paper, C denotes some positive (generally large) constant and λ denotes some positive (generally small) constant, where both C and λ may take different values in different places. A B means there exists a constant C > 0 such that A ≤ CB holds uniformly. A ∼ B means A B and B A. For the multi-indices α = (α 1 , α 2 , α 3 ) and β = ( ξ3 . Similarly, the notation ∂ α will be used when β = 0, and likewise for ∂ β . The length of α is denoted by |α| = α 1 + α 2 + α 3 . β ≤ α means that β j ≤ α j for each j = 1, 2, 3, and α < β means that β ≤ α and |β| < |α|. For notational simplicity, let ·, · denote the L 2 inner product in R 3 ξ with the L 2 norm | · | 2 , and let (·, ·) denote the L 2 inner product either in R 3 x × R 3 ξ or in R 3 x with the L 2 norm · . Moreover, we define |g| 2 ν = ν(ξ)g, g , g 2 ν = (ν(ξ)g, g). For an integer m ≥ 0, we use H m to denote the usual Sobolev space. We also define the space Z q = L 2 (R 3 ξ ; L q (R 3 x )) for q ≥ 1 with the norm For an integrable function g : R 3 → R, its Fourier transform g = F g is defined by for k ∈ R 3 , where i = √ −1 ∈ C is the imaginary unit. For two complex vectors a, b ∈ C 3 , (a|b) = a · b denotes the dot product over the complex filed, where b is the complex conjugate of b.
For q ∈ R, the velocity weight function w q = w q (ξ) is always denoted by with ξ = (1 + |ξ| 2 ) 1/2 . For an integer N and l ≥ N , we define the instant energy functional Remark 1.1. The analysis here can be used to deal with the case when ǫ = ǫ(t) > 0 and similar results can also be obtained provided that (q − γ) 2 ǫ(t) ≤ δ i hold for i = 0, 1 and every t ≥ 0. This means that for the Fokker-Planck-Boltzmann equation (1.1) with ǫ ≡ 0 and κ > 0, i.e.
we can use the scaling used in [23] to transform the above problem into (1.1) with ǫ = κ = κ(1 + 3κt) −1 and similar results can also be obtained provided that (q − γ) 2 ǫ(t) = (q − γ) 2 κ(1 + 3κt) −1 ≤ δ i hold for i = 0, 1 and every t ≥ 0. It is easy to see that a sufficient condition to guarantee the validity of the above inequalities is that κ > 0 is sufficiently small as imposed in [23] and it is worth to pointing out that when γ → 1 − and by taking q = 1, one can see that the assumptions (q − γ) 2 κ(1 + 3κt) −1 ≤ δ i hold even without the smallness restriction on κ. In such a sense, our result generalizes the result obtained in [23] even for the hard sphere intermolecular interaction.
Remark 1.2. It is worth to point out that here we use the weight function w l−|β| q to capture the term |ξ||∂ β u| generated by the ξ-derivatives ∂ β acting on the Fokker-Planck operator in term of the weaker dissipation rate ∂ β u ν . Remark 1.3. The rates of convergence are optimal under the corresponding assumptions in the sense that they coincide with those rates given in (4.1) at the level of linearization.
There have been a lot of studies on the Fokker-Planck-Boltzmann equation (1.1). DiPerna and Lions [5] proved the global existence of the renormalized solutions for the Cauchy problem (1.1) and (1.3). Hamdache [20] obtained the global existence near the vacuum state in terms of a direct construction. It is shown in [23] that a strong solution of the equation (1.1) for initial data near the global Maxwellian exists globally in time and tends asymptotically to another time-dependent self-similar Maxwellian in the large-time limit for the hard sphere case (1.4) with γ = 1. Li and Matsumura in [23] first introduced an appropriate scaling to transform (1.1) with ǫ ≡ 0 and κ > 0 into (1.1) with ǫ = κ → κ(1 + 3κt) −1 and then achieved their goals by employing the pioneering L 2 energy method based on macro-micro decomposition around a local Maxwellian developed for the Boltzmann equation [24], [25]. For the case −1 ≤ γ ≤ 1, the long time behavior to the Cauchy problem of (1.1), (1.3) is studied by constructing the compensating functions to this system, while the main goal of this paper is to obtain the global existence of classical solutions for (1.1) and (1.3) and the corresponding optimal time decay of the solutions under Grad's angular cut-off assumption for the whole range of intermolecular interaction −3 < γ ≤ 1.
In the perturbation theory of the Boltzmann equation for the global well-posedness of solutions around global Maxwelians, the energy method was first developed independently in [25,24] and in [16,18]. We also mention the pioneering work [32] and its recent improvement [33] which are based on the spectral analysis and the contraction mapping principle. We remark that the energy method based on macro-micro decomposition around a local Maxwellian [23] for the Fokker-Planck-Boltzmann eqution for the hard sphere case does not apply to the problem under our consideration with −3 < γ < 1. Our approach is based on the methods in [11,12] for the Vlasov-Poisson-Boltzmann system. For more information related to the Boltzmann equation and the kinetic theory, the reader can also refer to [4,3,13,30] and references therein.
Before concluding this section, we sketch main ideas used in deducing our results. One of the main difficulties lies in the fact that the dissipation of the linearized Boltzmann operator L for non hard-sphere potentials can not control the full nonlinear dynamics due to the velocity growth effect of |ξ||∂ β u| generated by the ξ-derivatives ∂ β acting on the Fokker-Planck operator. A suitable application of a weight function w l−|β| q can indeed yield a satisfactory global existence of classical solution to the Fokker-Planck-Boltzmann equation for the case −2 ≤ γ ≤ 1, while for the very soft potential case −3 < γ < −2, we cannot close our energy estimate by only employing the coercivity of the linearized collision L as for the case of −2 ≤ γ ≤ 1. Still and all, we can combine both the coercivity of L and L F P and divide the integral domain about ξ into two parts: the first part {ξ| ξ ≤ R} can be control by the coercivity of L with the smallness of ǫ while the second part {ξ| ξ > R} by the coercivity of L F P when we choose R large enough.
The time rate of convergence to equilibrium is an important topic in the mathematical theory of the physical world. As pointed out in [31], the exist general structures in which the interaction between a conservative part and a degenerate dissipative part lead to the convergence to equilibrium, where this property was called hypocoercivity. Here, indeed, we provide a concrete example of hypocoercivity property for the nonlinear Fokker-Planck-Boltzmann equation in the framework of perturbation. We employ the methods developing by Duan and Strain [9,10]. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case energy estimates with the help of the proper Lyaponov-type inequalities lead to the optimal time-decay rate of perturbed solution under some conditions on initial data. As in [12], unlike the periodic domain [29], the main difficult of the deducing the decay rates of solution for the soft potential is caused by the lack of spectral gap for the linearized collision operator L. We need a more delicate estimate on the time decay of solution to the corresponding linearized equation in the case of the whole space R 3 based on the weighted energy estimates, a time-frequency analysis method, and the construction of some interactive energy functionals. We also mention that Zhang and Li [36] have obtained the similar decay rate for the case −1 ≤ γ ≤ 0 by employing the compensating function which is different from us.
The rest of this paper is arranged as follows. We prove the global existence of solutions to the perturbed problem by establishing the a priori energy estimates on the microscopic and macroscopic dissipations which are derived in Sections 2 and 3, respectively. In the last section, we devote ourselves to obtaining the optimal temporal decay estimates of the global solutions for both the hard potentials and the soft potentials.

Macroscopic dissipation
In this section, we will obtain the macroscopic dissipation rate To this end, we shall first apply the macro-micro decomposition (1.8) to the equation (1.5) to discover the macroscopic balance laws satisfied by (a, b, c). Multiply (1.1) by the collision invariants 1, ξ and |ξ| 2 to find the local balance laws As in [9], define the high-order moment functions To obtain the second system of macroscopic equations, we split u = Pu + {I − P}u to decompose the equation Applying A jm (·) and B j (·) to both sides of (2.4), and using Now we focus on the macroscopic equations (2.3) and (2.6) to estimate the higher order derivatives of the macroscopic coefficients (a, b, c) in L 2 norm. For this purpose, we first give a lemma without proofs. Roughly speaking, the idea is just based on the fact that the velocity-coordinate projector is bounded uniformly in t and x, and the velocity polynomials and velocity derivatives can be absorbed by the global Maxwellian M which exponentially decays in ξ.
Lemma 2.1. For any |α| ≤ N and 1 ≤ j, m ≤ 3, it holds that Moreover, for any |α| ≤ N − 1 and 1 ≤ j, m ≤ 3, it holds that Next we state the key estimates on the macroscopic dissipation in the following theorem.
is the linear combination of the following terms over |α| ≤ N − 1 and 1 ≤ j ≤ 3: Proof.
Step 2. Estimate on c. For any η > 0, it holds that d dt (2.17) Indeed, applying ∂ α x with |α| ≤ N − 1 to the macroscopic equation (2.6) 3 , multiplying it by ∂ j ∂ α x c and then integrating it over R 3 , we have (2.18) Use (2.3) 3 to replace ∂ t c and estimate I c 1 as (2.21) Here we used the conservation of mass (2.3) 1 . Take summation (2.21) over |α| ≤ N − 1 to get (2.22) Step 4. Combination. We have finished the estimates of a, b, c. With them in hand, let us multiply (2.12) and (2.17) by a constant M > 0 and take summation of both of them as well as (2.22). One can first choose M > 0 sufficiently large such that the first term on the right-hand side of (2.22) can be absorbed by the dissipation of b and c. By fixing M > 0, one can choose η > 0 sufficiently small such that the first terms on the right-hand side of (2.12) and (2.17) are absorbed by the full dissipation of b and c. Hence, we have proved (2.11). Cauchy's inequality and (2.7) yield which implies (2.10). Therefore one has finished the proof of Theorem 2.1.

Global Existence
In this section, we shall devote ourselves to obtaining the existence of classical solutions to (1.5) globally in time. For this purpose, we first collect some estimates for the linearized Fokker-Planck operator L F P and the collision operators L and Γ.
For the linearized Fokker-Planck operator L F P , we have the following two results. The first one is concerned with the dissipative property of the linearized Fokker-Planck operator L F P without weight  [7]) L F P is a linear self-adjoint operator with respect to the duality induced by the L 2 ξ -scalar product. Furthermore, there exists a constant λ F P > 0 such that For the dissipative property of the linearized Fokker-Planck operator L F P with the weight w l−|β| q , we have Lemma 3.2. It holds that for any l ≥ 0, Proof. Integrating by parts yields for each R > 0. Here, we have used the fact that We estimate the terms on the right hand side of (3.3). First, If ξ is bounded, then ξ −2 ∼ ν(ξ) which implies For the corresponding weighed estimates on the linearized Boltzmann collision operator L and the nonlinear collision operator Γ, we have Lemma 3.3. ( [19], [17]) Consider the inverse power law with −3 < γ ≤ 1. If η > 0 and m ≥ 0, then there are C η , C > 0, such that Lemma 3.4. It holds that for any l ≥ 0, Next, as the first step, we shall obtain the dissipation rate To this end, we consider the non-weighted energy estimates on the solution u of (1.5)-(1.6). Taking ∂ α x of the equation (1.5) yields Applying (1.7), (3.1) and (3.8) with l = 0 to (3.10), we thus get the following lemma.
Lemma 3.5. It holds that for each t > 0, (3.11) For the second step, we consider the weighted energy estimates on u to get the dissipation rate Lemma 3.6. There is a positive constant δ 0 such that if Proof.
Step 4. Combination. First, let us multiply (3.11) by a constant M 1 > 0 and sum it with (2.11). Note that it holds that (2.10) and Thus, one can take M 1 > 0 such that the terms on the right-hand side of (2.11) can be absorbed and In the further linear combination one can take M 2 > 0 large enough to absorb all the dissipation terms on the right-hand sides of (3.14), (3.17) and (3.19), which implies  6) can be proved in terms of the energy functional E q,l (u)(t) given by (1.10), and the details are omitted for simplicity, see [16,17,23] with a little modification. Now we have obtained the unform-in-time estimate (3.13) over 0 ≤ t ≤ T with 0 < T ≤ ∞. By the standard continuity argument, the global existence follows provided the initial energy functional E(u 0 ) is sufficiently small.

The hard potential case
In this subsection, we devote ourselves to obtaining the time decay rate of the global solution u to the Fokker-Planck-Boltzmann equation (1.5)-(1.6) in the hard potential case (0 ≤ γ ≤ 1). For this purpose, we first deduce some estimates for the Cauchy problem: where u 0 (x, ξ) and G = G(t, x, ξ) with PG = 0 are given. Formally, the solution u to the Cauchy problem (4.1) can be written as the mild form where e tB denotes the solution operator to the Cauchy problem of (4.1) with G ≡ 0. We first show that the operator e tB has the proposed algebraic decay properties as time tends to infinity. The idea of the proofs is to make energy estimates for pointwise time t and frequency variable k, which corresponds to the spatial variable x.
Proof. Estimate on b. We claim that for 0 < η < 1, it holds that (4.5) In fact, the Fourier transform of (2.13) gives We then take the complex inner product with b j to find (4.6) which implies |A jm (R)| 2 (1 + |k| 2 )|{I − P} u| 2 ν . Thus, I 1 is bounded by (4.7) For I 2 , using the Fourier transform of (2.3) 2 to replace ∂ t b j , we have (4.9) Therefore, one can take the real part of (4.6) and plug the estimates (4.7) and (4.9) into it to discover (4.6). Estimate on c. For any 0 < η < 1, we have (4.10) In fact, multiply the Fourier transform of (2.6) 3 (4.11) For I 4 , using the Fourier transform of (2.3) 3 to replace ∂ t c, one has (4.12) Hence, (4.10) follows by taking the real part and applying the estimates of (4.11) and (4.12), and then taking the summation over 1 ≤ j ≤ 3. Estimate on a. We claim that it holds for any 0 ≤ η < 1 that (4.13) In fact, using (4.8), and taking the complex inner product with ik j a, and then taking the summation over 1 ≤ j ≤ 3, one has (4.14) The first there terms on the right-hand side of (4.14) are bounded by while for the last term, it holds that Here we used the Fourier transform of (2.3) 1 : Then, one can deduce (4.13) by putting the above estimates into (4.14) and taking the real part. Therefore, (4.4) follows from the proper linear combination of (4.5), (4.10) and (4.13) by taking M > 0 large enough and 0 < η < 1 small enough. Note that This completes the proof of lemma 4.1.
Lemma 4.2. κ 1 > 0 exists such that E( u)(t, k), which is defined by for any t ≥ 0 and k ∈ R 3 .

The soft potential case
In this subsection, we shall obtain the time decay of the solution u to the Cauchy problem (1.5)-(1.6) in the soft potential case (−3 < γ < 0). For this, we first establish the time decay of the evolution operator e tB , which is stated as follows.