Well-posedness of Cauchy problem for Landau equation in critical Besov space

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.


1.
Introduction and main result. The Landau equation is a fundamental model in kinetic theory that describes the evolution of the density of particles in a plasma. In this work, we consider the spatially inhomogeneous Landau equation, which is given by where f = f (t, x, v) ≥ 0 is the density of particles on position x ∈ R 3 and with velocity v ∈ R 3 at time t ≥ 0. The collision operator Q L is a bilinear operator acting only on the velocity variable v and reads as where we used the usual shorthand . The matrix-valued function a(v) = (a i,j (v)) 1≤ i,j≤3 is non-negative, symmetric and depends on the interaction between particles, which is usually assumed by where I = I 3×3 is the unit matrix on R 3 and v ⊗ v = (v i v j ) 1≤ i,j≤3 . One calls hard potentials if γ ∈ (0, 1], Maxwellian molecules if γ = 0, soft potentials if γ ∈ (−3, 0) and Coulombian potential if γ = −3. One also divides the soft potentials into two categories: moderately soft potentials if γ ∈ (−2, 0) and very soft potentials if γ ∈ (−3, −2]. In this paper, we are interested in the Cauchy problem (1) with Maxwellian molecules, since the Landau operator enjoys very nice spectral property in that case.
In what follows, we are concerned with the Landau equation around the absolute Gaussian distribution in R 3 : We can linearize it with the perturbation x, v).
Then it follows that ∂ t g + v · ∇ x g + Lg = L(g, g), with f 0 = µ(v) + √ µ(v)g 0 , where L(g) = L 1 (g) + L 2 (g) with and We observe (see for example, [23]) that the linear operator L is non-negative and its null space has dimension 5, which is given by P denotes the orthogonal projector onto the null space N . Furthermore, in case of Maxwellian molecules, one has the explicit form where is the harmonic oscillator and is the Laplace-Beltrami operator on the unit sphere S 2 and P k (k = 1, 2) is the orthogonal projection onto the Hermite basis. Landau equation is a fundamental equation to describe collisions among charged particles interacting with their Coulombic force. There are lots of known results concerning the well-posedness and large-time behavior of solutions to the Landau equation.
In spatially homogeneous case, Villani [33] constructed weak solutions for the Coulombic interaction with γ = −3 up to some defect measures. Subsequently, in [32], he extended the result to the Landau equation without the presence of spatial dependence. In the Maxwellian molecules case γ = 0, Villani proved an exponential in time convergence to equilibrium. For the hard potential γ ∈ (0, 1], Desvillettes and Villani [16,17] investigated the existence, uniqueness and smoothness of classical solutions. They proved a functional inequality for entropy-entropy dissipation that is not linear, from which the polynomial in time convergence of solutions towards equilibrium was also shown. Recently, Carrapatoso [9] proved the optimal exponential decay to equilibrium with the decay rate given by the spectral gap of the associated linearized operator, by using the method developed by Gualdani, Mischler and Mouhot [20]. Morimoto and Xu [28] proved the ultra-analytic effect for the Cauchy problem of linear Landau equation in the case of γ = 0. Li and Xu [26] studied the nonlinear Landau operator by introducing the spectral analysis.
In spatially inhomogeneous case, Guo [21] proved the global-in-time existence of classical solutions to the Landau equation in a period box. Later, Hsiao-Yu [22] extended Guo's results [21] to the whole space. Baranger and Mouhot [7] studied the explicit spectral gap estimates to the linearized Landau operator with hard potentials. Mouhot [30] established the coercivity estimates for a general class of interactions including hard potentials and soft potentials. Lerner-Morimoto-Starov-Xu [23] showed that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian.
To the best of our knowledge, there are various studies concerning the wellposedness of solutions to the Boltzmann and Landau equation, see for example, [1,2,3,8,16,17,25,27,31,32] and references therein. Very recently, Duan, Liu and the last author of this paper [19] first introduced the Chemin-Lerner type spaces involving the microscopic velocity and established the global existence of strong solutions near Maxwellian for the cut-off Boltzmann equation. Subsequently, Morimoto and Sakamoto [29] extended their result to the non-cutoff Boltzmann equation by using the triple norm that was introduced by Alexandre-Morimoto-Ukai-Xu-Yang [2,5]. However, there are few results concerning the global existence for the Landau equation in spatially critical Besov spaces. So it is very interesting to work a result for (1), since the collision operator between the Boltzmann equation and Landau equation are fundamentally different. As a first step, by using the spectral analysis on the nonlinear Landau operator, we investigate the Cauchy problem (1) with Maxwellian molecules (γ = 0). The main result is now stated as follows. .
Above norms of the Chemin-Lerner space will be rigorously defined by Appendix B. The Chemin-Lerner space without involving the microscopic velocity was initiated by [12] to establish the global existence of solutions to the incompressible Navier-Stokes equations. Observe that the regularity index s = 3/2 that the Besov space is subjected to B 3/2 2,1 (R 3 ) → L ∞ (R 3 ), but the Sobolev space H 3/2 (R 3 ) is not embedded into L ∞ , it thus is critical for the algebra with respect to the spatial variable. Also, we remark that the Chemin-Lerner space L ∞ t L 2 v (B 3/2 2,1 ) enjoys stronger topology than the usual mixed space L ∞ t L 2 v (B 3/2 2,1 ), which allows to get a good control for nonlinear collision terms. The proof of Theorem 1.1 is separated into several parts. Regarding the local-in-time existence part, we employ the Picard's iteration scheme to prove the local existence of solution. As a matter of fact, solving the linear Landau equation is quite elaborate, which will be presented by Appendix A for clarity. To get round the difficulty that the dual space of L ∞ T L 2 v (B 3/2 2,1 ) is unknown, firstly, we try to find a weak solution in the wider space L ∞ ([0, T ]; L 2 (R 6 x,v )) based on the Hahn-Banach extension theorem. Secondly, using various commutators estimates (which are well developed in Section 4) to get the desired solution in L ∞ T L 2 v (B 3/2 2,1 ). In the part of global-in-time existence, we establish those trilinear estimates to the nonlinear Landau collision operator, which play a key role achieving the global solution. For that end, our proof heavily depends on spectral properties of Landau operator with Maxwellian molecules (see Section 2 for details) in contrast with [19,29]. Finally, the standard continuity argument enables us to obtain Theorem 1.1. Remark 1. The Bony's para-product decomposition have been widely used in the study of fluid dynamics, see for example [10,11,13,14,15] and references therein, however, there are few results available that one applies the Besov space theory to the global existence of kinetic equations. The recent works [19,29] are devoted to the Boltzmann equation. We would like to mention that Theorem 1.1 should be our first effort, and the research of Landau equation in cases of hard potentials and soft potentials, which is left in near future.
The paper is arranged as follows. To make the manuscript self-contained, in Section 2, we recall the spectral analysis properties and some key estimates of Landau collision operator. In Section 3, we establish some crucial estimates for nonlinear estimates, for instance, trilinear estimates for Landau operator. Section 4 is devoted to establish commutator estimates for Landau operator. In Section 5, we prove the global existence of solution to the Landau equation. To do this, we prove the local existence of solutions to (1) by using the iteration Scheme and construct a priori estimates arising from the coercivity of linear Landau operator. However, we cannot deduce the dissipative estimate for the macroscopic part Pg directly. To overcome the difficulty, as in [19,29], we shall perform the macro-micro decomposition and deduce a fluid dynamics system of macroscopic projection of g. Consequently, by using the standard energy method, we can obtain the estimate on the macroscopic dissipation. Appendix A is dedicated to the solvability of linear Landau equation, which is based on the duality argument and Hahn-Banach extension theorem. Some definitions of Chemin-Lerner type spaces and some inequalities of Besov spaces used in this paper are collected in Appendix B.
2. Preliminary: Analysis of Landau collision operator. For convenience of reader, we present the spectral properties for Landau operator briefly, see [8,23,25,26] for more details. First of all, one has an explicit expression for the linearized Landau operator with Maxwellian molecules.
where L 1 and L 2 are equal to Here and below, ∆ S d−1 stands for the Laplace-Beltrami operator on the unit sphere S d−1 and P k (k = 1, 2) is the orthogonal projection onto the Hermite basis.
There is the algebra property of nonlinear Landau operators on the basis {ϕ n,l,m } (see [25,26] and referencein), for n, l ∈ N, m ∈ Z, |m| ≤ l, let's denote where Γ( · ) is the standard Gamma function, and -L (α) n is the Laguerre polynomial of order α and degree n, The notations σ = (cos θ, sin θ cos φ, sin θ sin φ), N l,m are the normalisation factor and P |m| l is the Legendre functions of the first kind of order l and degree |m| From [23,24], we see that those spherical harmonics are equivalent to the real spherical harmonics Y m l (σ) for l ≥ 0 and −l ≤ m ≤ l, which are defined by Y 0 0 (σ) = 1 √ 4π and for any l ≥ 1, and the eigenfunctions {ϕ 1,1,m1 , |m 1 | ≤ 1} are given by Next, we present the algebraic property of nonlinear Landau operator.
Proposition 3. For the coefficients of the Proposition 1 defined in (3), we have i) For n, l ∈ N, n ≥ 1, ii) For n, l ∈ N, l ≥ 1, Proof. It follows from (3) that A + n,l−1,m,m1 = 4(l + 1) We recall again that, for σ, κ ∈ S 2 , By using the fact that, and the orthogonal of {Y −m * l (ω), l ∈ N, |m * | ≤ l} on S 2 , one can verify that Hence, we arrive at (7). On the other hand, we have By employing the fact with l − 1, where l ≥ 1 that, This leads to the inequality (8).
3. Nonlinear estimates of Landau collision operator. In order to prove Theorem 1.1, we establish nonlinear estimates of Landau collision operator.

Trilinear estimates.
The orthogonal projectors {S N , N ∈ N} and {S N , N ∈ N} are defined as follows, where g n,l,m = g, ϕ n,l,m and then we have Let us begin with bounding the nonlinear term (L(f, g), h) L 2 .
where the self-adjoint operator L is defined by Bounding J 4 is similar, so we arrive at Next, we turn to estimate J 2 . Precisely, By exchanging the order in the last summation, we deduce from (7) in Proposition 3 that |m1|≤1,|m|≤ l+1 |m+m1|≤ l A − n−1,l+1,m,m1 h n,l,m1+m Then we obtain Those terms J 3 , J 5 , J 6 and J 7 may be treated along the same line as J 2 with aid of (4), (5), (6) in Proposition 2 and (8) in Proposition 3. Therefore, we can deduce that Together with those estimates on J 1 -J 7 , we get Similarly, when estimating the terms J 1 -J 7 , taking L ∞ norm on the position variable x for f and taking L 2 norm on the position variable x for g, we obtain Combining (10)- (11) gives the desired inequality (9). Hence, the proof of Theorem 3.1 is finished.
Furthermore, optimal information will be obtained if splitting the functions into frequency packets of comparable sizes. Indeed, one has Theorem 3.2. Let f, g, h ∈ S(R 6 x,v ). It holds that for any j ≥ −1.
Proof. For f, g, h ∈ S(R 6 x,v ), it follows from Proposition 1 and the orthogonal property of {ϕ n,l,m ; n, l ∈ N, |m| ≤ l}, that x,v ≤ I 1 + I 2 + I 3 + I 4 + I 5 + I 6 + I 7 with I 1 = 2n+l≥ 0 |m|≤ l (2(2n + l) + l(l + 1)) (∆ j (f 0,0,0 g n,l,m ) , ∆ j h n,l,m ) L 2 |m|≤ l,|m2|≤2 |m+m2|≤ l A 2 n−1,l,m,m2 (∆ j (f 0,2,m2 g n−1,l,m ) , ∆ j h n,l,m+m2 ) L 2 x , To estimate I 1 , Bony's decomposition comes into play in our context. The product of u and v can be decomposed into The above operators T and R are called "paraproduct" and "remainder", respectively. Moreover, it follows from Lemma 7.1 that Consequently, we are led to the inequality x . Furthermore, with aid of Lemma 7.2, we arrive at The Cauchy-Schwarz inequality enables us to get Similar to the proof of the term I 1 , one can verify that We now estimate the remaining terms I 2 , I 3 , I 5 , I 6 , I 7 . The process of the proofs with respect to these five terms are almost the same, so we can take I 5 as the example.
Regarding the term I 51 , we use Cauchy-Schwarz inequality again and get where the last summation can be estimates by (4) in Proposition 2, |m2|≤2,|m|≤ l+2 |m+m2|≤ l A 1 n−2,l+2,m,m2 ∆ j h n,l,m+m2 Then it follows that 16n(n − 1) 3 Bounding I 52 and I 53 essentially follows from the same procedure as I 51 , so we get Therefore, by combining those estimates, we conclude that Similarly, we can get the following estimates for I 2 , I 3 , I 6 , I 7 : Putting above estimate of I 1 -I 7 together, we eventually conclude that (12).
By using Theorem 3.2, we establish crucial estimates for Landau collision operator in the framework of Besov space, which are used to achieve the global-in-time existence.
x, v) be three suitably functions, then it holds that, Proof. It follows from Theorem 3.2 and Cauchy-Schwarz inequality that By changing the order of the summation, we have fulfills c(p) 1 ≤ 1. Hence, by Fubini's theorem and Young's inequality, we get We exchange the order in the summation that It is obviously that, for s > 0, then it follows that .
Consequently, we end the proof of (13).

Coercivity of linear Landau operator.
In order to obtain energy estimates, the coercivity of linear operator which indicates the microscopic dissipation plays a key role. The lower estimate of (L 1 g, g) and the upper estimate of (L 2 f, g) are shown in the following theorem.
Theorem 3.4. For the linear operators L 1 and L 2 , it holds that for some positive constant C.
Proof. By Proposition 1, we have (2(2n + l) + l(l + 1)) ϕ n,l,m g n,l,m , Furthermore, we see that Thus, keeping in mind that (L(g), g) , we can obtain (14) with aid of Young's inequality. The proof of Theorem 3.4 is completed.
Moreover, we have the direct consequence of Theorem 3.4. Corollary 1. For the linear operators L 1 and L 2 , it holds that for s > 0 and 0 < T ≤ ∞.

Macro projections of nonlinear operator.
For the nonlinear Landau operator L(g, g), we have the following macro projections.
Proposition 4. Let ϕ n,l,m be the set of eigenfunctions. For g, h ∈ S(R 3 ), we have

HONGMEI CAO, HAO-GUANG LI, CHAO-JIANG XU AND JIANG XU
Proof. For temperate functions f, g ∈ S(R 3 ), it follows from Proposition 1 that We need to compute A + 0,0,0,m1 , A − 0,1,m,m1 for |m| ≤ 1 and |m 1 | ≤ 1. The direct computation shows that Therefore, if we take the case of f = g, then we obtain which is just (15). Now we prove the equality (16). For |m| ≤ 2, It is not difficult to find that Substituting into the above formula, we get On the other hand, by a direct computation, we obtain Consequently, we deduce that Hence, the proof of Proposition 4 is completed.
Based on Proposition 4, we have the following estimate in spatially Besov spaces.
Proposition 5. Let s > 0 and φ(v) be the finite combination of the eigenfunctions Then it holds that for any T > 0.

Commutators estimates.
To improve the regularity of weak solution, we need delicate commutator estimates involving the nonlinear term L(f, g) and cut-off functions with respect to variables x and v. Notice that In addition, we have which leads to and

Commutators with moments. Let
for 0 < δ < 1. Here and below, we agree with the norm Proof. It follows from (20) that HONGMEI CAO, HAO-GUANG LI, CHAO-JIANG XU AND JIANG XU which implies that Therefore, we arrive at In the following, A 1 , · · · , A 5 can be estimated one by one. For A 1 , by using integration by parts and Cauchy-Schwarz inequality, we have where we used the fact Next, we proceed the similar procedures for A 2 -A 5 and obtain Hence, (22) follows from (23)-(24) directly.

Commutators with a mollifier in the x variable.
Inspired by [1,29], we can obtain the commutator of the collision operator with a mollifier in the x variable.
, which is just (25) and where we used Theorem 3.1 and K δ L 1 Proof. By using the nonlinear term L (f, g) in (20) and integration by parts, we have Owing to the fact furthermore, one can deduce that where we agree with ∂ l = ∂ v l . The direct calculation enables us to get It is not difficult to see that where the last two derivations are constants. Consequently, we are led to So, by Cauchy-Schwarz inequality, we conclude that

HONGMEI CAO, HAO-GUANG LI, CHAO-JIANG XU AND JIANG XU
where the following estimates are used: . By employing the similar calculations as B 1 , we have which gives Integration by parts with respect to the variable v * , we arrive at Putting the above estimates for B 1 -B 4 together, we achieve (26) eventually.
Based on Propositions 6-8, we obtain commutator estimates involving the nonlinear term L and various cut-off functions.
Proof. Note that For J 1 , it follows from Proposition 6 that x,v ) . For J 2 , thanks to Proposition 7 and Young's inequality, we get for > 0, where C is some constant depending only on . For J 3 , by Proposition 8, we obtain x ) . The first inequality in Proposition 9 follows from above estimates for J 1 , J 2 and J 3 . In addition, we can get the second inequality according to Theorem 3.1.

Commutators for linear operator.
Moreover, we also need some commutators with the linear operator L 1 .
Proposition 10. For 0 < δ < 1, it holds that Proof. By using Lemma 2.1, we have For the term D 1 , integration by parts allows to get Furthermore, we obtain On the other hand, it follows from the direct calculation that where we used the relation We obtain (27) directly. Therefore, the proof of Proposition 10 is completed.
Proposition 11. For 0 < δ ≤ 1, it holds that Proof. From Lemma 2.1, we obtain For E 1 , by the direct calculation, we arrive at which indicates that The inequality (28) is followed directly. Hence, the proof of Proposition 11 is completed.
As a consequence of Propositions 10-11, we get the commutator estimate.
Proposition 12. Let g ∈ S(R 6 x,v ). There exist some positive constants C > 0 independent of T > 0 such that where > 0 is sufficiently small.
Proof. Obviously, we see that For F 1 , it follows from the Proposition 10 that x,v )) , where we used (21). For F 2 , thanks to Proposition 11, we obtain x,v )) . Therefore, we finish the proof of Proposition 12.

5.
The existence of solution. In this section, we are devoted to obtain the global existence of the solution to the inhomogeneous Landau equation.

The local-in-time existence.
We first establish the local-in-time existence of solution to the Cauchy problem (2).
2,1 ). Proof. Firstly, we construct the following sequence of iterating approximate solutions: starting from g 0 (t, x, v) ≡ g 0 (x, v). Taking g = g n+1 , f = g n and T = min{T 0 , 1/(4C 2 0 )} in Theorem 6.1 gives g n where ε 0 > 0 is chosen such that 2C 0 ε 0 ≤ ε. Secondly, it suffices to prove the convergence of the sequence {g n } in the space Y , which is defined by

LANDAU EQUATION 861
Set w n = g n+1 − g n . It follows from (29) that ∂ t w n + v · ∇ x w n + L 1 w n = L(g n , w n ) + L(w n−1 , g n ) − L 2 w n−1 with w n | t=0 = 0. By employing the similar energy estimate leading to (55), we get for some 0 < λ < 1. Clearly, we see that {g n } is a Cauchy sequence in Y , so there is some limit function g ∈ Y such that g n → g as n → ∞. The standard procedure enables us to know that g is the desired solution to the Cauchy problem (2) satisfying The non-negativity of the solution. For the solution of the Cauchy problem (2) that obtained in Proposition 13, which is the limit of the sequence of (29), coming back to the original Landau equation, it is also the limit of a sequence constructed successively by the following linear Cauchy problem Then, the non-negativity of the solution to the Cauchy problem (1) can be proved by the same methods as in [4,21].

The global-in-time solution.
We prove now that we can extend the above local-in-time solution to a globalin-time solution, which heavily depends on the key a priori estimate. For this end, we define the energy functional and the dissipation functional 2,1 ) , respectively. Furthermore, it is shown that Proposition 14. Let g ∈ Y be the solution to the Cauchy problem (2). It holds that for any T > 0, where C > 0 is some constant independent of T .
Based on the above global a priori estimate (30) and the local existence result (Proposition 13), Theorem 1.1 is followed by the standard continuity argument. We feel free to skip the procedure. The interested reader is referred to [19] for similar details.
We split the proof of Proposition 14 into several parts.

Estimate on the macroscopic dissipation.
In this part, we bound the macroscopic dissipation arising from Landau collision operator. We by P denote the projection operator on ker N , which is given by In terms of the macro-micro decomposition, the distribution function g(t, v, x) can be decomposed as g = Pg + (I − P)g.
Precisely, the macroscopic dissipation of g is included in the following proposition.

Proposition 15. It holds that
for any T > 0.
With aid of the orthogonal of {ϕ n,l,m }, we infer that (a, b, c) which is the coefficient of the macroscopic component Pg (31) satisfies the fluid-type system where i, j = 1, 2, 3. Applying the cut-off operator ∆ p with p ≥ −1 to the system (33) implies that As in [18,19], we denote the temporal interactive functionals as follows: where 0 < κ 2 < κ 1 1 are some constants (to be confirmed below).
By multiplying the fifth equality of (34) by ∂ xi ∆ p c and summing up on i with 1 ≤ i ≤ 3, one can get

HONGMEI CAO, HAO-GUANG LI, CHAO-JIANG XU AND JIANG XU
Using the third equality of (34) and Young's inequality, we arrive at for 01 > 0, where C 01 is a constant depending on 01 . Furthermore, by using Young's equality again, we are led to for λ 1 > 0 and 01 , 02 > 0, where C 02 and C 01, 02 are some constants depending on 01 , 02 .
Multiplying the fourth equality of (34) by ∂ xi ∆ p b j + ∂ xj ∆ p b i and summing up 1 ≤ i, j ≤ 3, we obtain Then, substitute the second equality of (34) to eliminate ∆ p ∂ t b and get x for 11 > 0, where C 11 is a constant depending on 11 . Consequently, there exist some constant λ 2 > 0 such that the following inequality holds where C 11 and C 11, 12 are positive constants depending on 11 , 12 > 0. Multiplying the second equality by ∂ xi ∆ p a and summing up 1 ≤ i ≤ 3, by similar calculations as above, we get for λ 3 > 0, where C 21 is a positive constant depending on 21 . Put above energy estimates together and choose 01 , 11 , κ 1 , κ 2 small enough. Consequently, there exists a positive constant λ > 0 such that Integrating the above inequality with respect to t over [0, T ] and taking square roots on both sides give Multiplying the above inequality by 2 p 2 and then taking the summation over p ≥ −1 to get Clearly, it is not difficult to check that By using Proposition 5 (taking s = 1 2 ) and the Sobolev embedding B where we have used the following estimates: By inserting (36)-(37) into (35), we conclude that , which is just (32).

5.4.
Estimate on the nonlinear term L(g, g).
Proposition 16. Let g = g(t, x, v) be suitably smooth function. It holds that Proof. With aid of the macro-micro decomposition, we split L(g, g) into four terms: .

Note thatḂ
, Proposition 21 and Lemma 7.7, we arrive at In fact, other collision terms can be estimated at a similar way. Precisely, Therefore, combine those estimates to finish the proof of Proposition 16.
Proof of Proposition 14.
Proof. Applying ∆ p (p ≥ −1) to (2) and taking the inner product with ∆ p g over where we used (v · ∆ p ∇ x g, ∆ p g) L 2 x = 0. Integrate the above inequality with respect to the time variable over [0, t] with 0 ≤ t ≤ T and then take the square root of both sides of the resulting inequality. Consequently, we obtain , which implies that (by using Proposition 16) It follows from Proposition 15 that (performing the calculation α × (32) + (38) in fact) that 2,1 ) + ( E T (g) + E T (g))D T (g), which leads to (30) directly if taking α > 0 sufficiently small. 6. Appendix A. This section can be regarded as an independent one in regard to the present paper, which is devoted to the local existence in spatially critical Besov spaces for linearized Landau equation.
admits a weak solution g ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) satisfying Due to the fact that the dual space of L ∞ T L 2 v (B 3/2 2,1 ) is unknown, the proof of Theorem 6.1 is a little bit complicated. For clarity, we divided it into several parts.

The local existence of weak solution.
Firstly, we establish the local existence of weak solution to the Cauchy problem (39) by using the duality argument and Hahn-Banach extension theorem.
then the Cauchy problem (39) admits a weak solution x,v )). Proof. The strategy of that proof was originated from [5], and well developed by [29]. We consider the joint operator where (·) * is taken with respect to the scalar product in Furthermore, we arrive at In the following, we consider the vector subspace x,v )). Indeed, the above inclusion is true due to similar calculations in Theorem 3.1. For g ∈ L 2 x,v , we get For g 0 ∈ L 2 (R 6 x,v ), we define the linear functional as follows where h ∈ C ∞ ([0, T ], S(R 6 x,v )) with h(T ) = 0. It follows from (41) that the operator G is injective. The linear functional Q is hence well-defined. We obtain Hence, Q is a continuous linear form on (U, · L 1 ([0,T ],L 2 x,v ) ). By using the Hahn-Banach theorem, Q can be extended as a continuous linear form on L 1 ([0, T ]; L 2 (R 6 x,v )). It follows that there exists g ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) satisfying It implies that for all h ∈ C ∞ 0 ((−∞, T ), S(R 6 x,v )), Therefore, g ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) is a weak solution of the Cauchy problem (39). The proof of Proposition 17 is completed.
In next steps, we need to improve the regularity of weak solution g ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) in both velocity and position variables.

Regularity of weak solution in velocity variable.
To do this, we smooth out the function f . Set f N = S N f for N ∈ N. Then, we have f N ∈ L ∞ T L 2 v (H ∞ x ) and the following property.
x . For the left hand side of (42), it can be obtained from Lemma 7.5. For the right hand side, we have 2,1 ) and we used the estimate ≤ 1 |p|≤1 2 ps 1 c(p) 1 < +∞. Therefore, the proof of Lemma 6.2 is completed.
Then, according to the commutator estimate in Section 4, we have the following estimate for the weak solution.
admits a weak solution g N (t, x, v) ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) satisfying Proof. It follows from Proposition 17 that the Cauchy problem (43) admits a weak solution In what follows, we show (44) under the assumption that f N L ∞ ([0,T ]×R 3 Let 0 < δ, δ < 1. We use a weighted function W δ (v) = δ v −2 and mollifiers M δ (D v ), S δ (D x ) defined as in Section 4. Taking the inner products of (43) 4 and integrating the resulting equation with respect to the time t ∈ [0, T ] and (x, v) ∈ R 6 . We obtain To the term v · ∇ x g N , we bound it as x ) . On the other hand, regarding linear terms L 1 , L 2 , we obtain, by Proposition 12 and Lemma 3.4, where > 0 is sufficiently small. Combining (45)-(46), it is shown that is small enough. Taking T sufficiently small (for example, CT < 1 4 ) and passing to the limit δ → 0, we get x,v ) , where we used the fact L and T are both small, we obtain x,v ) for 0 < δ < 1. Now letting δ → 0 and taking square root to the resulting inequality give the desired (44) for a weak solution g N ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )). Hence, the proof of Proposition 18 is finished.

Regularity of weak solution in position variable.
In the following, we need to obtain the regularity of g N with respect to the position variable x.
for any κ > 0, where and C N > 0 is a constant depending only on N .
Proof. Due to L where C κ > 0 is a constant depending only on κ > 0.
By using the Lemma 3.2 and Bony's decomposition, we divide the inner product into three terms: For the term H 1 , noticing that we have Hence, it follows that , where we used Lemmas 7.2, 7.5 and 7.7 and the following estimate p≥−1 |j−p|≤4 For H 2 , we get Finally, H 3 can be estimated as follows Together with above three inequalities, we achieve (47). This ends the proof of Lemma 6.3.
Based on Proposition 18 and Lemma 6.3, we can obtain the regularity of the weak solution g N to the Cauchy problem (48) and get the corresponding energy estimate dependent of N . Proposition 19. There exist 2 > 0 and T 0 > 0 such that for all where C N > 0 is a constant depending on N and f N = S N f .
Proof. We consider a weak solution g N ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) to the above Cauchy problem (48). To do this, applying ∆ p (p ≥ −1) to (48) and taking the inner product with ∆ p g N over R 3 x,v ), where we used Lemma 3.4 and Corollary 1. Integrating the above inequality with respect to the time variable over [0, t] with 0 ≤ t ≤ T and taking the square root of both sides of the resulting inequality. Multiplying the resulting inequality by x . Taking supremum over 0 ≤ t ≤ T on the left side and summing up over p ≥ −1, we obtain , where we used the Proposition 6.2 and Lemma 6.3 because the weak solution g N satisfies the Proposition 18. Then, for the small constant T > 0 and the small norm x . Letting κ → 0, we deduce that x , which ends the proof of Proposition 19.

Energy estimates in Besov space.
From (49) in Proposition 19, we see that g N satisfies 2,1 ) < +∞. That is, the regularity of the weak solution g N has been improved. However, the upper bound for g N depends on N . In the following, we perform a technical procedure to eliminate the dependence of N .
Proof. The proof is similar to that of Lemma 6.3. With aid of Bony's decomposition, we have Thus H 3 can be estimated as follows: . Therefore, we can obtain (50).
Based on Lemma 6.4, we obtain the energy estimate for the weak solution g N , which is independent of N .
admits a weak solution g N ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) satisfying g N L ∞ where C > 0 is a constant independent of N .
Proof. We consider the weak solution g N ∈ L ∞ ([0, T ]; L 2 (R 6 x,v )) to (51). Applying ∆ p (p ≥ −1) to (51) and taking the inner product with ∆ p g N over R 3 x,v ), where we used Lemma 3.4 and Corollary 1. Integrating the above inequality with respect to the time variable over [0, t] with 0 ≤ t ≤ T and taking the square root of both sides of the resulting inequality. Multiplying the resulting inequality by 2 x . Take supremum over 0 ≤ t ≤ T on the left side and sum up over p ≥ −1, we obtain Applying ∆ p (p ≥ −1) to (53) and taking the inner product with 2 3p ∆ p w M,M over Integrating (54)   Next, we prove (40). Applying ∆ p (p ≥ −1) to (39), taking the inner product with 2 3p ∆ p g over R 3 x × R 3 v and using Lemma 3.4 and Corollary 1, we get d dt 2 3p ∆ p g 2 L 2 x,v + 2 3p L 1 2 ∆ p g 2 L 2 x ≤ 2 3p+1 (∆ p L(f, g), ∆ p g) L 2 x,v + 2 3p C( ∆ p S 2 f 2 L 2 x,v + ∆ p g 2 L 2 x,v ).
Integrating (56) with respect to the time variable over [0, t] with 0 ≤ t ≤ T , taking the square root of both sides of the resulting inequality and summing up over p ≥ −1. It follows from Lemma 3.3 that , C √ T < 1 4 ), we get the desired inequality (40).
7. Appendix B. For convenience of reader, we recall the Littlewood-Paley decomposition and definition of Besov spaces. The reader is also referred to [6] for more details.
It is also convenient to introduce the low-frequency cut-off: The Littlewood-Paley decomposition is "almost" orthogonal.
Lemma 7.1. For any u ∈ S (R d ) and v ∈ S (R d ), the following properties hold: Additionally, it is crucial that we have Lemma 7.2. Let 1 ≤ p ≤ ∞ and u ∈ L p x , then there exists a constant C > 0 independent of p, q and u such that ∆ q u L p x ≤ C u L p x , S q u L p x ≤ C u L p x . Now, we turn to the definition of the main functional spaces and norms in the present paper. For the distribution f = f (t, v, x), we define the Banach space x ))) for 0 < T ≤ ∞, 1 ≤ p 1 , p 2 , p 3 ≤ ∞, where the norm is given by with the usual convention if p 1 , p 2 , p 3 = ∞.
Next, we present the definition of the Chemin-Lerner type space, which were initialed in [12]. In addition, we also use the weighted Sobolev spaces H s (R 6 x,v ). For s, ∈ R, one define H s (R 6 x,v ) = g ∈ S (R 6 x,v ); v g ∈ H s (R 6 x,v ) , where the weight is with respect to the velocity variable v ∈ R 3 .