Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off

In this article we study the large-time behavior of perturbative classical solutions to the Fokker-Planck-Boltzmann equation for non-cutoff hard potentials. When the initial data is a small pertubation of an equilibrium state, global existence and temporal decay estimates of classical solutions are established.


1.
Introduction and main result. The Fokker-Planck-Boltzmann equation is concerned with the motion of particles in a thermal bath where the bilinear interaction is one of the main characters [2,4]. In the present paper, we consider the initial value problem of the Fokker-Planck-Boltzmann equation where F (t, x, v), (t, x, v) ∈ R + × R 3 × R 3 is the distribution function and , κ are given nonnegative constants. The bilinear collision operator Q(F, G) is given by where F = F (t, x, v ), F = F (t, x, v), G * = G(t, x, v * ) and G * = G(t, x, v * ) for short, and the pairs (v, v * ) and (v , v * ) stand respectively for the velocities of particles before and after collision, with the following momentum and energy conservation rules fulfilled, v + v * = v + v * , |v | 2 + |v * | 2 = |v| 2 + |v * | 2 .
From the above relations we have the so-called σ-representation, with σ ∈ S 2 ,    The cross-section B(v−v * , σ) in (1.2) depends on the relative velocity |v−v * | and the deviation angle θ with cos θ = σ ·(v −v * )/|v −v * |. Without loss of generality we may assume that B(v − v * , σ) is supported on the set θ ∈ (0, π/2] where (v − v * ) · σ ≥ 0, since as usual B can be replaced by its symmetrized version, and furthermore we may suppose it takes the following form: where γ ∈ (−3, 1]. In this paper, we assume that γ + 2s ≥ 0, = κ > 0 and the angular function b(cos θ) is not locally integrable: for c b > 0 it satisfies c b θ 1+2s ≤ sin θ b(cos θ) ≤ 1 c b θ 1+2s , s ∈ (0, 1). In what follows, we consider the classical solutions to (1.1) near a global Maxwellian µ(v) = (2π) −3/2 e −|v| 2 /2 . To this end, we use f to denote the perturbation of F around the Maxwellian µ as F = µ + √ µf, then the fluctuation f satisfies the Cauchy problem Here, the linear operator L, the bilinear form Γ(f, g) and the classical linear Fokker-Planck operator L F P are respectively given by , It is well konw that L is non-negative and the null space N of L is given by (1.5) Now we define P the orthogonal projection from L 2 (R 3 v ) to N. As in [15], for any given function f , one can write (1.6) Therefore, we have the macro-micro decomposition introduced in [15], where I denotes the identity operator, Pf and {I − P}f are called the macroscopic component and the microscopic component of f respectively. As in [9,11], for later use, one can write P as      Pf = P 0 f ⊕ P 1 f, P 0 f = a(t, x)µ 1/2 , P 1 f = {b(t, x) · v + c(t, x)(|v| 2 − 3)}µ 1/2 . (1.8) Before stating our main results, we list some notations as follows. Throughout this paper, c 1 , c 2 , · · · stand for various generic positive constants. C denotes some positive constant(generally large) and λ denotes some positive constant(generally small), where both C and λ may take different values in different places. Furthermore A B means A ≤ CB, and A B means B A. In addition, A ≈ B means A B and B A. The multi-indices α = (α 1 , α 2 , α 3 ) and β = (β 1 , β 2 , β 3 ) will be used to record spatial and velocity derivatives respectively. Specifically, v3 . Similarly, the notation ∂ α will be used when β = 0, and likewise for ∂ β . The length of α is |α| = α 1 + α 2 + α 3 . α ≤ α means that each component of α is not greater than that of α, and α < α means that α ≤ α and |α | < |α|. Let ·, · denote the standard L 2 inner product in R 3 v , with its L 2 norm given by | · | 2 . L 2 l denotes the weighted space with the norm, for l ∈ R |f | 2 For later use, We now list series of notations introduced in [13,14]. Let S (R 3 ) be the space of the tempered distribution functions. N s,γ denotes the weighted geometric fractional Sobolev space Here the anisotropic metric d(v, v ) measuring the fractional differentiation effects is given by and 1 A is the standard indicator function of the set A. In R 3 ×R 3 , we use ||·|| N s,γ = || | · | N s,γ || L 2 (R 3 x ) . For an integrable function f : R 3 → R, its Fourier transform is defined bŷ For two complex vectors a, b ∈ C 3 , (a|b) = a ·b denotes the dot product over the complex field, whereb is the ordinary complex conjugate of b.
For r ≥ 1, we define the mixed Lebesgue space As in [13], the velocity weight function w = w(v) always denotes w(v) = v . For l ∈ R, the velocity-weighted | · | N s,γ -norm is given by We also define B C ⊂ R 3 to be the Euclidean ball of radius C centered at the origin and denote L 2 (B C ) as the space L 2 on this ball and likewise for other spaces.
For an integer K and K ≥ 4. Fix l ≥ 0. Given a solution f (t, x, v) to the Boltzmann equation (1.4), we define an instant energy functional E K,l (t), which satisfies 9) and the dissipation rate (1.10) Our main result are stated as follows.
x, v) be the solution to the Cauchy problem (1.4) obtained in Theorem 1.1. If K,l and are sufficiently small. Then for any t ≥ 0 we have where K,l is defined by There has been much rigorous analysis established recently about the Fokker-Planck-Boltzmann equation (1.4). DiPerna and Lions [5] obtained the global existence of the renormalized solutions in the L 1 framework. Hamdache [19] proved the global existence near the vacuum state by a direct constuction. In [20], Li and Matsumura showed that a strong solution exsits globally in time and tends time-asymptotically to an self-similar Maxwellian with respect to time and velocity for a small perturbation of an absolute Maxwellian. Under Grad's angular cut-off assumption, Xiong, wang and Wang [26] established the global classical solution existence and time decay estimates to the equation (1.4) in the perturbation framwork.
Recently there have been many important results on the global existence and long time behavior for the Boltzmann equation and other kinetic equations. For the hard-sphere model with angular cutoff, the global existence of solutions to the Vlasov-Poisson-Boltzmann system was proved in [27] and [8] in different function spaces, and the corresponding large-time bahavior of solutions were obtained in [28] and [9], respectively. For the hard potential 0 < γ < 1 or hard sphere, we can refer to [6,15,16,21,22]. For the soft potentials −3 < γ < 0, there also has been much important progress. Guo [17] used energy methods developed in [16,18] to generalize the results in [3] to the case −3 < γ < 0 and obtained the time decay rate of e −λt p for some λ > 0 and 0 < p < 1 in [25]. For −1 < γ < 0, Ukai and Asano established the global solutions to the Boltzmann equation near a global Maxwellian and the decay rate of t −α with 0 < α < 1, their optimal case in whole space is α = 3/4. Without the angular cut-off assumption, Strain [23] obtained the optimal time decay rates of classical solutions to the Boltzmann equation with hard and soft potentials in the whole space, Duan and Liu [12] established the global existence and convergence rates of classical solutions to the Vlasov-Poisson-Boltzmann system when intial data is near Maxwellians. Our approach is based on the methods in [10,11] for the Vlasov-Poisson-Boltzmann system, although their results are in the angular cut-off case.
In this paper, we establish the global classical solution to equation (1.4) by the standard energy method, and obtain the same decay rate as theirs in [26], which is studied using Fourier analysis, the only difference is that the equation is in the context of non-cutoff cross-section. One of the main difficulty lies in the the term v |∂ β {I − P}f | generated by the v-derivatives ∂ β acting on the Fokker-Planck operator, for β ∈ R 3 . We use the velocity weight function w l−|β| to control the above term v |∂ β {I − P}f |, where 0 ≤ |β | < |β|, with help of the smallness of , we can close our energy estimate. Detailed procedures can be illustrated by the following proofs.
The rest of this paper is arranged as follows. In Section 2, we give several lemmas for late use frequently. In Section 3 and Section 4, we give the proof of the global classical solution to the equation (1.4) and the decay rate respectively.

2.
Preliminaries. In this section, we give several lemmas which will be frequently used later.
The first lemma is concerned with the lower bound for the linearized collsion operator L.
Lemma 2.1 (Theorem 8.1 in [13]). There exists a constant δ 0 > 0 such that The second lemma concerns the coercive interpolation inequalites for L.
Lemma 2.2 (Lemma 2.6 in [13]). For any multi-indices α, β, l ≥ 0, and any small η > 0 there is a positive constant C η such that Furthermore, there exists some constant C > 0 such that Remark 1. The estimate (2.3) indeed holds for any l ∈ R as follows from Lemma 2.4 and Lemma 2.5 in [13].
The following trilinear estimates are the main ones we use below.
is a non-negative constant which is derived from the Leibniz rule. Also, Γ β is the bilinear operator with derivatives on the Maxwellian µ given by For this Γ β , we have Lemma 2.4 (Proposition 6.1 in [13]). For any multi-index β ∈ R 3 , any l ∈ R. Suppose that φ are the smooth rapidly decaying basis vectors in (1.5). Then Lemma 2.5 ((6.6) in [13]). Let γ +2s ≥ 0. For any multi-index β, In what follows, we shall write down the macroscopic equations of equation (1.4) by applying the macro-micro decomposition (1.7). For this purpose, we define the high-order moment functions Then, as in [9], one can deduce from (1.4) a fluid-type system of equations Now we give some estimates to the higher order functions (2.8) in L 2 norm.
Lemma 2.6. Let γ + 2s ≥ 0, 1 ≤ j, m ≤ 3. For any |α| ≤ K, we have (2.12) Furthermore, for any |α| ≤ K − 1, it holds that (2.13) Proof. Since the velocity can be absorbed by the global Maxwellian µ, which exponentially decays in v, it is straightforward to estimate (2.12). Now we estimate (2.13). It is easy to confirm that: Recall that Lg = −Γ( √ µ, g) − Γ(g, √ µ), using (2.4), we have With the above estimates, we get The other one shares similar arguments above. (2.14) Proof. We just need to prove one of them above. Recall that We konw that Fistly we estimate (2.14) for the third term Γ(f, {I − P}f ) above. Use Leibniz formular in (2.5) and apply Lemma 2.5( herein we let , whichever contains the minimal total number of derivatives, and use the Sobolev embedding for this term to obtain (2.14). For the second term in (2.15), we notice that where the ψ i (t, x) are the elements from (1.6) and the φ i are the velocity basis vectors in (1.5). Thus from Lemma 2.4 Here |[a, b, c]| is just the Euclidean square norm of the coefficients from (1.4). Use the same arguments above to get (2.14).

LVQIAO LIU AND HAO WANG
The last case to consider is Γ(Pf, Pf ). Applying Lemma 2.4 again to get (2.14). However, when |α| = 0, we combine the L 2 * (R 3 x ) gradient Sobolev inequality, for 2 * = 6 is the Sobolev conjugate of 2, with With the estimates to the higher order functions above, we state the key estimates on the macroscopic dissipation in the following Proposition.
where E int (t) is the linear combination of the following terms over |α| ≤ K − 1 and 1 ≤ j ≤ 3 : Applying (2.12) and the above estimate, we can easily deduce (2.17).
step 2. Estimate on c. For any η > 0, it holds that d dt In fact, for |α| ≤ K − 1 , applying ∂ α to the macroscopic equation (2.10) 3 , multiplying it by ∂ xj ∂ α c and then integrating it over R 3 (2.23) For I c 1 , use (2.9) 3 to replace ∂ t c to obtain Using (2.13)and (2.14), we compute Therefore, (2.22) follows by plugging (2.24) and (2.25) into (2.23), and summing it over 1 ≤ j ≤ 3 and |α| ≤ K − 1. step 3. Estimate on a. Let |α| ≤ K − 1. Apply ∂ α to (2.9) 2 , multiply it by ∂ α ∇ x a and then integrate it over R 3 x to get x ) , where we have used (2.9) 1 . Using (2.12) and taking summation the above estimate over |α| ≤ K − 1, we have We choose M sufficiently large such that the first term on the right hand side of (2.26) can be absorbed by the dissipation of b and c. Fixing this M > 0, then we choose η > 0 sufficiently small such that the first terms on the right hand side of (2.19) and (2.22) are absorbed by the full dissipation of b and c. Thus, we have proved (2.18).
Lastly, for the macroscopic components, we have two results. Note that in (1.6), we easily get the following lemma, the details are omitted for simplicity.

Remark 2.
Recall that E int (t) in Proposition 1. With (2.18), (2.27) and (2.28), we get the important estimate to be used in the next section (2.29) 3. global existence. The following short time existence of classical solution to (1.4) can be established by performing the standard arguments as in [13,20,17].
Here we omit the details.
Here, G(f (t)) is defined by

LVQIAO LIU AND HAO WANG
Based on the above local existence result, we can establish the global existence for equation (1.4). Now we give some estimates for the linearized Fokker-Planck operator L F P and the collision operators Γ.
The first one is concerned with the dissipative property of the Fokker-Planck operator L F P without weight, which has been proved in [7,1].
Lemma 3.2. L F P is a linear self-adjoint operator with respect to the duality induced by the L 2 v -scalar product. Furthermore, there exsits a constant λ F P > 0 such that The second lemma concerns the dissipative property of the Fokker-Planck operator L F P with weight.
Proof. Integrating by parts yields Here, we have used γ + 2s ≥ 0 and the fact that Applying Lemma 3.2 and the above estimate, the lemma is proved.
The following Lemma is concerned with weighted estimates on the nonlinear collision operator Γ. Using (3.1), (3.5) in [23], we have where E K (t) = E K,0 (t) and D K (t) is given by Furthermore, for any l ≥ 0, Now we start to establish the global energy estimate for (1.4). Fistly, we consider the unweighted estimate on the solution f of (1.4).
Proof. Taking the spatial derivatives of ∂ α of (1.4) to obtain 1 2 Then with Lemma 2.1, Lemma 3.2 and the above estimates, we easily get (3.6).
Next, we consider the weighted energy estimates. Before stating the next lemma, we set Here λ > 0 may depend on l.

LVQIAO LIU AND HAO WANG
Then with Cauchy-Schwartz inequality, we easily get |Γ 2 | D 0 K (t). Apply Lemma 2.2, we see that, for some appropriate constant λ 1 > 0, Furthermore, from Lemma 3.3, we get Plug the above estimates into (3.9) and choose small enough to achieve Step 2. For 1 ≤ |α| ≤ K, we take ∂ α of (1.4), multiply the result by w 2l ∂ α f for l ≥ 0, and then integrate in x, v to achieve Using the similar arguments above, we easily compute Then we obtain (3.12) Step 3. Fix |α| + |β| ≤ K with |β| ≥ 1. Applying ∂ α β to (3.8), we have where Then mutiplying (3.13) by w 2l−2|β| ∂ α β {I − P}f and integrate over R 3 × R 3 to get 1 2 whereĨ We begin to estimateĨ 2 , for a small constant η > 0, Lastly considering the first term on the right side of (3.14), we have With the above estimate and Lemma 3.3, we have
Proof of Theorem 1.1. We choose the initial data E K,l (0) such that }. Recall that C 0 , C 1 , λ are constants in (3.1) and (3.7), and δ appears in Lemma 3.1. Note that E K,l (0) ≤ M ≤ δ, then from Lemma (3.1) there exists a solution f for some T > 0, and from the local estimate (3.1), we have E K,l (t) ≤ C 0 M for 0 < t < T . We define Note that on [0, t] for 0 < t < T * , E K,l (t) ≤ C 0 M < 1, then E K,l (t) ≤ E K,l (t). For the choice of M , λ − 2C 1 √ C 0 M > 0, the global energy estimate (3.7) implies This induces that T * = +∞. Thus we finish the proof of Theorem 1.1.

Time decay.
This section is devoted to obtaining the time decay rate of the global solution f to the Fokker-Planck-Boltzmann equation (1.4). Firstly, we consider the linearized Fokker-Planck-Boltzmann equation For the nonlinear system (1.4), the non-homogeneous source term is given by In this case g = {I − P}g. Formally, the solution f to the Cauchy problem (4.1) can be written as the mild form where e −tB is the linear solution operator for the Cauchy problem to (4.1) with g = 0.
The first lemma is concerned with the estimate on the macroscopic dissipation.
Lemma 4.1. For any t ≥ 0 and k ∈ R 3 , there is M > 0 such that the free energy functional E f ree (t, k) defined by and

(4.6)
Here e is a linear combination of {e m }, which are the smooth exponentially decaying velocity basis vectors contained in (1.6) and (2.8).
Then we can deduce (4.16) by putting the above estimates into (4.17) and taking the real part.
Next we give estimates on the microscopic dissipation and microscopic weighted dissipation, which will be used below.
Apply Remark 1 to achieve, for any l ∈ R, Using the same arguments in Lemma 3.3 yields For the last term J 1 , owing to the rapid decay in the coefficients of (1.6) 1 , we get |J 1 | ≤ η|{I − P}f | 2 N s,γ l + C η |k| 2 |Pf | 2 2 . Plug the above three estimates into (4.19) and chose a small > 0 to obtain for any t ≥ 0.
Collecting the above estimates as well as (4.28) gives (4.26).