The Boltzmann equation with frictional force for very soft potentials in the whole space

We develop a general energy method for proving the optimal time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space, for both hard potentials and soft potentials. With the help of this method, we establish the global existence and temporal convergence rates of solution near a given global Maxwellian to the Cauchy problem on the Boltzmann equation with frictional force for very soft potentials i.e. \begin{document}$ -3 .

1. Introduction and main results.
1.1. The problem. This paper is concerned with the following Boltzmann equation with external force proportional to the macroscopic velocity u(t, x) in the whole space R 3 : with initial data F (0, x, v) = F 0 (x, v).
(2) Here F = F (t, x, v) ≥ 0 stands for the velocity distribution functions for the particles with position x = (x 1 , x 2 , x 3 ) ∈ R 3 and velocity v = (v 1 , v 2 , v 3 ) ∈ R 3 at time t ≥ 0 and the term ζu represents the frictional force which is proportional to the macroscopic velocity u(t, x) and we can normalize the positive constant ζ to be 1

YINGZHE FAN AND YUANJIE LEI
without loss of generality. The bilinear collision operator Q(F, G) acting only on the velocity variable is defined by where in terms of velocities v * and v before the collision, velocities v and v * after the collision are defined by which follow from the conservation of momentum and kinetic energy during the collision process The function |v − v * | γ q 0 (θ) in (3) is the cross-section depending only on cos θ = (v − v * ) · ω/|v − v * | and |v − v * | , and it satisfies −3 < γ ≤ 1 and it is assumed to satisfy the Grad's angular cutoff assumption 0 ≤ q 0 (θ) ≤ C| cos θ|.
The exponent γ is determined by potential of intermolecular forces, which is called the hard potential when 0 ≤ γ ≤ 1 including the Maxwell model γ = 0 and the hard sphere model γ = 1, q 0 (θ) = C| cos θ|, and the soft potential when −3 < γ < 0, especially, the case −2 ≤ γ < 0 is called the moderately soft potential and −3 < γ < −2 very soft potential, cf. [1,2,3,16]. The Boltzmann equation with frictional force describes the motion of the rarefied gas with friction force on the microscopic aspect. The corresponding compressible Euler system with frictional damping describes the macroscopic motion of the fluid. As is well known, there is a close relationship between the two equations. For example, the first-order approximation of the Boltzmann equation with frictional force by Hilbert expansion is the compressible Euler system with frictional damping formally when the number of Knudsen is small. Also for the compressible Euler equation with frictional damping, Hsiao-Liu [22] has shown that large time behavior of its global solutions can be described by the compressible flow through porous media. Since there are many open problems in Euler system with frictional damping, We can use the properties of the solution of the microscopic equation i.e. Boltzmann equation with frictional force to study the properties of the macroscopic equation, especially for the large time of the solution. So the well-posed of the solution to the Boltzmann equation with frictional force becomes very meaningful.
In the following text, we will solve this problem, and in the proving procedure, we will provide a general method for proving the time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space, which conclude both hard potentials and soft potentials.
To this end, if we set then the Cauchy problem (1)-(2) can be reformulated as with given initial data Here 2 g denote the linearized and nonlinear collision operators, respectively. It is well-known, cf. [17], that the null space of L is Pf and {I − P}f are called the macroscopic component and the microscopic component of f (t, x, v), respectively. Under the Grad's angular cutoff assumption (4), L is non-negative and coercive in the sense that there is a constant κ 0 > 0 such that holds for f = f (t, x, v). Furthermore, L can be written as L = ν − K, where ν denotes the collision frequency and K is a velocity integral operator with a real symmetric integral kernel K(v, v * ). The explicit representation of ν and K will be given in the next section. Notations.
• 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.
, respectively, and L 2 x,v , L 2 x , L 2 v are used for the case when m = 0. ·, · denotes the L 2 inner product in R 3 v , with the L 2 norm | · | or | · | 2 . And (·, ·) denotes the L 2 inner product in R 3

YINGZHE FAN AND YUANJIE LEI
• For q ≥ 1, we also define the mixed velocity-space Lebesgue space ball of radius C centered at the origin, and L 2 (B C ) stands for the space L 2 over B C and likewise for other spaces.

1.2.
Main results. Before giving our main results, we introduce the following mixed time-velocity weight function where 0 < q 1, |β| ≤ l, 0 < ϑ ≤ 1 4 , which had been applied in [12,23] to get the extra dissipative terms by the energy estimates such as: which can control the following terms: if and only if −2 ≤ γ < 0 and u has a better temporal decay estimate. The reference [23] gives a detailed proof. However, when −3 < γ < −2, the above strategy can not hold any more! To overcome the above difficulties, we introduce the following general weight function w −|β|,κ (t, v) where the precise range of the parameter ϑ will be specified later, also define several temporal energy functionals which stands for the energy of the gas molecules: where N ≥ 0 is an integer, and ≥ N is a constant. Meanwhile the energy dissipation rate functional D (j) ,N,κ (t) corresponding to E (j) ,N,κ (t) is defined as is sufficiently small, there exists a unique global solution f (t, x, v) to the Cauchy problem where the energy functional E (j) lj ,N,−γ (t) is defined in (13), σ n,j is defined as:

Remark 1.
• It is worth pointing that we use the weighted functional e q v (1+t) ϑ in (15) instead of e q v 2 (1+t) ϑ in [12], the advantage is that the requirement of • In our proof, we provide a general method for proving the optimal time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space for both hard potentials and soft potentials. This method is different from that in [21]. Now we present the main ideas in the proof as the following three folds: (i). First of all, take w −|β|,κ with κ = −γ, as the above discussion, when −3 < γ < −2, unlike (11), we estimates the terms as follows: With the help of linear decay analysis and Duhumel principle, if we suppose the energy function E by mathematical induction, where the χ is the characteristic function. If we take l j−1 = l j + 2 and suppose the energy function E (ii). Secondly, unlike (11) and (17), we take w −|β|,κ with κ = 1, = l * where l * is an undetermined constant which is strictly greater than −γ l 0 + 5 2 , since and where we use the time decay property (18). However, for the transport term v · ∇ x f , Since γ 2 < − γ 2 − 1 < 1 2 holds for all −3 < γ < −2, it does not lead to the increase of the weight if we neglect the factor (1 + t) −1−ϑ in the extra dissipative term (10). Therefore, if we set different time increase rate σ n,j for |α|+|β|=n,|β|=j Based on the above argument, if we assume that E (iii). Finally, in order to guarantee the smallness of E (0) l0+ 5 2 ,N,−γ (t), we split it into macroscopic part and microscopic part, i.e.
With the help of the above estimates, finally, we will construct the a priori estimate and close it. The global solvability result follows. Our manuscript is organized as follows. In Section 2, we will give some key estimates, which include the coercivity property on the linear operator and the L 2 −estimates on the nonlinear terms. In section 3, we will show that the three energy functionals enjoy time decay rates, time increase rates and bounded, respectively. Then we deduce the desired priori estimates and close it. Finally, we complete the proof of Theorem 1.1.

Preliminaries. Recall that
We list in the following lemma velocity weighted estimates on the collision frequency ν(v) and integral operator K with respect to the velocity weighted function (12) In this section, we cite some fundamental results concerning the weighted energy type estimates on the linearized collision operator L and the nonlinear term Γ, whose proofs can be found in [12,30]. The first lemma concerns the linearized operator L. Lemma 2.1. (cf. [12]) Let 0 < κ < 3, ∈ R, and 0 ≤ q ≤ 1, If |β| > 0, and for any η > 0, there is C η > 0 such that Furthermore, if |β| ≥ 0, then for any η > 0, there is C η > 0 such that The following two lemmas concern the estimates on nonlinear term Γ. The second lemma is concerned with the corresponding weighted estimates on the nonlinear term Γ. For this purpose, similar to that of [26], we can get that where g i = g i (t, x, v) (i = 1, 2) and the summations are taken for all β 0 + β 1 + β 2 = β, α 1 + α 2 = α. From Lemma 3 in [26], one can deduce that x, v), β 0 + β 1 + β 2 = β and α 1 + α 2 = α, we have the following results: or Furthermore, it holds that and hold for any 0 < t ≤ T .
Then the evolution operator satisfies for any t > 0.
Proof. The proof is similar with Theorem 3.1 of [11], here we omit it for brevity.
The following lemma concerns the nonlinear estimates.
Proof. For example, for j ≥ 1, ,N,−γ (t) (35) follows by the similar way. Thus we have completed the proof of this lemma.
3. The proofs of main results. This section is devoted to proving our main results based on the continuation argument. Local existence for the Cauchy problem (5)-(6) in certain weighted Sobolev space is now well-understood, cf. [20], thus we will get the global existence with the standard continuity argument if we can close a global priori estimate in the same weighted Sobolev space in which the local solution is constructed.
To make the presentation clear, we divide the rest of this section into three subsections. The first one focuses on deducing the temporal time decay of the energy functional on f (t, x, v). The second is to control the energy functionalẼ l * ,N,1 (s) with the weight function w l * −|β|,1 (t, v) which includes the time increasement rate.
The third is to prove the boundness of the energy functional E (0) l0+ 5 2 ,N,−γ (t) with the weight function w l0+ 5 2 −|β|,−γ (t, v). The last is to close the a priori estimate and complete the proof of Theorem 1.1.

The temporal time decay rates of solution on f (t, x, v). This subsection concentrates on obtaining the Lyapunov inequality for
holds for any 0 ≤ t ≤ T .
Proof. To this end, we rewrite (5) as

YINGZHE FAN AND YUANJIE LEI
We divide the proof into three steps: Step 1. Take summation over j ≤ |α| ≤ N , with the help of Lemma 2.5, 2.6, it holds that Step 2. Energy estimates with the wight function w −|β|,−γ (t, v): Let us write down the time evolution equation of {I − P}f : Firstly, multiplying (41) by w 2 ,−γ (t, v)∂ α {I − P}f with |α| = j. and integrating it over R 3 x × R 3 v , and applying Lemma 2.5, 2.6, we have 1 2 Secondly, for the weighted estimates on the terms containing only x derivatives, we take ∂ α x to (39) 1 with j + 1 ≤ |α| ≤ N , then multiply it by w 2 ,−γ (t, v)∂ α f and integrate it over R 3 x × R 3 v , and we have d dt (43) Thirdly, for the weighted estimates on the mixed x−v derivatives: applying ∂ α β with 1 ≤ |β| ≤ N and |α| + |β| ≤ N to (41), multiplying it by w l−|β|,q (t, v)∂ α β {I − P}f and integrating over R 3 x × R 3 v , applying Lemma 2.5, 2.6, taking summation over {|β| = m, |α| + |β| ≤ N } for each given 1 ≤ m ≤ N and then taking the proper linear combination of those N − 1 estimates with properly chosen constants C m > 0 and η > 0 small enough, we have Step 3. In this step, we take a proper linear combination of the estimates (40),(64) (44) and(32). We can obtain that ,N,−γ (t) is given by where κ > 0 is small enough and C 1 and C 2 are chosen large sufficiently. It is easy to see thus we have completed the proof of the Lemma.
From the process of proof, in fact, we have not used the the exponential function e q v (1+t) ϑ of the weighted function (12).
The following lemma concerns the decay of the macro term of f (t). Proof. here and m can be chosen an enough large positive constant. By (30), one has N,N,−γ (s) and by Sobolev inequality, one also has N,N,−γ (s), then we arrive at (47) So we can obtain that where we have used the fact that Thus the proof of this lemma is complete. Now we are ready to deduce the temporal time decay rates of E (j) ,N,−γ (t). Lemma 3.3. For l j ≥ N , take l j−1 = l j + 2 , suppose the energy function E  (49) taking = l 0 and multiplying the above inequality by (1 + t) and is a small positive number. In a similar way, taking = l 0 + 1 2 in (50) and multiplying (50) by (1 + t) A proper linear combination of (51), (52) and (50) with = l 0 + 1 imply that Integrating the above inequality, one has (1 + t) which gives (49) with j = 0. For 1 ≤ j ≤ N − 1, similar to the energy estimates of E ,N,−γ (t), one also has d dt E by mathematical induction.If we take l j−1 = l j + 2 and suppose the energy function E (0) l0+ 5

Remark 2.
• From this lemma, we know that to establish the time decay rate of E (j) lj ,N,−γ (t), we should assume the functional E (0) l0+ 5 2 ,N,−γ (t) is sufficiently small.
• Indeed, combining Lemma 3.1, Lemma 3.2 with Lemma 3.3 gives a general method of proving the optimal time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space, for both hard potentials and soft potentials.
Secondly, for the weighted estimates on the terms containing only x derivatives, we take ∂ α x to (39) 1 with 1 ≤ |α| ≤ N , then multiply it by w 2 l * ,1 (t, v)∂ α f and integrate (63) Here we have used For the second term on the left hand of the above equation, For I 7 on the right hand of (63), we have from Lemma 2.1 Obviously, I 8 can be bounded by η ν 1 2 ∂ α f 2 + C η ∂ α u 2 . I 9 can be bounded by: For I 10 , Lemma 2.2 tells us that: and Consequently Collecting the above estimates, for 1 ≤ |α| ≤ N , one has Thirdly, for the weighted estimates on the mixed x − v derivatives: applying ∂ α β with |β| = j,1 ≤ j ≤ N and |α| + |β| = n, 1 ≤ n ≤ N to (57), multiplying it by

YINGZHE FAN AND YUANJIE LEI
Now we estimate term by term as follows. For the third term on the left hand of (65), we have For the first term I 11 on the right hand of (65), we have from Lemma 2.1 For the second term I 12 on the right hand of (65), we have It is straightforward to estimate I 13 and I 14 on the right hand of (65), the two terms are bounded by I 15 can be bounded by: Similar with I 10 , by applying Sobolev and Hölder inequalities, one has: To control the last term on the right hand side of the above inequality, we introduce (1 + t) −σn,j in which σ n,j − σ n,j−1 = 2(1+γ) γ−2 (1 + ϑ), 1 ≤ j ≤ n, σ n,0 = 0, then one can deduce that