VLASOV-POISSON-FOKKER-PLANCK IN FRACTIONAL SOBOLEV-LEBESGUE SPACES

A BSTRACT . In this paper, we are concerned with the local well-posedness of the Vlasov-Poisson-Fokker-Planck equation near vacuum in the fractional Sobolev-Lebesgue space for the large initial data. To achieve this goal, we mainly adopt the energy method. In order to obtain the energy estimate, we establish an L 2 - L q estimate related to the electronic term, and take advantage of the commutator estimates as well.


INTRODUCTION
In this paper, we study the following Vlasov-Poisson-Fokker-Planck equation (VPFP) in n-dimensional space: where f (t, x, v) denotes the distribution function of particles, x ∈ R n is the position, v ∈ R n is the velocity, t > 0 is the time, and n ≥ 5.
In statistical physics, the VPFP system is one of key equations governing the evolution of a distribution of particles over time. Specifically, it models the distribution of particles in a plasma with respect to position and velocity affected by gravitational or electrostatic forces, and the collision effects are produced by the Brownian motion of particles. In the stellar dynamical context, one of the fundamental problems is to incorporate, in the framework of a general theory, the effect of encounters between stars. Stellar encounters under Newtonian inverse square attractions influence the motion of starts in the manner of Brownian motion [4].
The Cauchy problem for the VPFP system has been studied for several decades. In 1984, Neunzert, Pulvirenti, and Triolo [12] proved the global existence of smooth solutions for the two-dimensional case using a probabilistic method. Two years later, Degond [5] proved the global existence for the VPFP system for dimensions only one and two with initial data in In the higher dimension, Victory [16] studied the VPFP system with initial data satisfying and Victory proved the existence of the global weak solutions under these conditions. Later, Rein and Weckler [14] proved the global existence of a classical solution in a three-dimensional case with initial condition in the following class: Then, Carrillo and Soler [2], proved the global existence of weak solutions for the VPFP system in threedimensions with the initial data in L 1 (R 2n ) ∩ L p (R 2n ), i.e. the initial data is still in L 1 -framework. They also studied the VPFP equations with measures in Morrey spaces as initial data, see [3]. All the literature mentioned above were concerned about the solutions in L 1 ∩ L ∞ . It comes naturally to ask how about in the L 2 framework? For instance, whether there is a solution in the fractional Sobolev space H s is an interesting question, which becomes the main theme in this paper. Besides, there is rare paper which devoted to the study of the solutions of the VPFP system in the fractional Sobolev space, which motivates us to take a step forward in this direction.
To study the well-posedness of (1.1), the main difficulty lies in estimating the electronic term ∇ x φ. Our strategy is to take advantage of the following type of commutator estimates related to the electronic term x , which is the basis for the estimate in the hybrid Sobolev-Lebesgue space. Combining with the L 2 -L q estimate with respect to ∇ x φ (see Section 3), we could then close the energy. Also, it is worth to mention that a weight w is necessary to be introduced to derive an L 2 -L q off-diagonal estimate which plays an important role to deal with the Poisson equation near vacuum.

PRELIMINARIES AND MAIN THEOREM
Before we state our main theorem, we would like to set our notations and definitions first.

Notations and definitions.
• Given a locally integrable function f, the maximal function M f is defined by where |B(x, δ)| is the volume of the ball of B(x, δ) with center x and radius δ. • Given f ∈ S Schwartz class, its Fourier transform Ff =f is defined bŷ and its inverse Fourier transform is defined by F −1 f (x) =f (−x). In this paper, we use F x f (x, v) to represent the Fourier transform of x only, and ξ to represent the dual variable of x.
• Let s > 0, the hybrid Sobolev-Lebesgue space with weight in v is defined bỹ where S is the dual space of Schwartz class. For the convenience, we useH s for the abbreviation ofH s (R 2n x,v ) whenever there is confusion arising. • A B means there exists a positive generic constant C independent of the main parameters such that A ≤ CB. A ∼ B means A B and B A.
Remark 2.1. The hybrid spaceH s possesses the different differential and integral properties on x and v variables.
2.2. Some useful lemmas. In this part, we collect some known results of the Riesz potential [1,15] and the boundedness of Hardy-Littlewood operator for later use. The pointwise estimate of the Riesz potential stated below is applied to derive the off-diagonal estimate in Section 3.

Lemma 2.2. ( [1])
For any multi-index ξ with |ξ| < α < n, there is a constant C such that for any f ∈ L p (R n ), 1 ≤ p < ∞, and almost every x, we have . Remark 2.3. In our paper, we consider −∆φ = R n f dv =: g, n ≥ 3. Thus, in our context, I α can be taken where ω n−1 = 2π n 2 Γ( n 2 ) is the (n − 1)−dimensional area of the unit sphere in R n . Additionally, we have the pointwise estimate The boundedness of the Riesz potential in Lebesgue space is needed in our proof as well.
where n > 4, c = c(p,q) and For more results of the Riesz potential and its applications in partial differential equations, see [6][7][8]. Now we give the boundedness of Hardy-Littlewood operator M which is defined by (2.1). i.e., The following commutator estimate is of importance in this paper, see Section 4. Lemma 2.6. ( [11]) Let 1 < p < ∞, and let 1 < p 1 , p 2 , p 3 , p 4 ≤ ∞ satisfy Then for any f, g ∈ S(R n ), the following holds Now we are ready to state our theorem.
Remark 2.8. In the definition of spaceH s , we only impose the differential assumption on the variable x, not on the variable v; by contrast, the weight function w(v) is only about v, not about x. So our space is different from the fractional Sobolev space, it is a hybrid Sobolev-Lebesgue space with weight.

L 2 -L q ESTIMATES
In this section, we are aiming to obtain an L 2 -L q estimates related to the electronic term ∇ x φ, which plays a fundamentally important role in the commutator estimate of the electronic term ∇ x φ.
First of all, we establish the boundedness of the solution of Laplacian equation in Lebesgue space.
Proof. Note that ∇ x φ = ∇ x (I 2 * g) by Lemma 2.4, therefore there holds and we applied Lemma 2.2 in the second line. On the one hand, the boundedness of Hardy-Littlewood operator Lemma 2.5 yields that On the other hand, by the boundedness of the Riesz potential Lemma 2.4, we have Consequently, , which ends the proof of this lemma.
Remark 3.2. This lemma explains the reason that we can not have a solution in the L 2 -framework for n ≤ 4, in some sense. Remark 3.3. We summarize the conditions imposed on the indices in Lemma 3.1 as follows: The following corollary will be used in our commutator estimate involving the electronic term ∇ x φ.

Corollary 3.4.
Assume −∆φ = g, then it holds that Proof. Observe that applying Lemma 3.1 with φ and g replaced by D s x φ and D s x g respectively yields the desired result.
Proof. Hölder's inequality leads to Note that w = v γ and γ > n, which implies that Thus, we ends the proof of Corollary 3.5.
Remark 3.6. ∇ x φ is a function of the variable x only, while f is a function of the variables x and v.H s is a hybrid Sobolev-Lebesgue space with weight depending on v only, which is defined in Section 2.1.
An L ∞ estimate is also needed in the proof of main result Theorem 2.7.

PROOF OF MAIN THEOREM
To prove Theorem 2.7, we split its proof into two parts which are existence and uniqueness of the solution to (1.1). Let us start with proving the existence. Proof of existence. In this part, we adopt the energy method and the iteration method. To do so, we need to close the energy by applying the commutator estimate. In this process, the L 2 -L q estimate of electronic term ∇ x φ plays a nice role.
Proof. We consider the following iterating sequence for solving the VPFP (1.1), Applying D s x to the first equation in (4.1), we have Multiplying (D s x f k+1 )w on both sides of (4.2), and then integrating over R n x × R n v yields that 1 2 We now estimate (4.3) term by term. For J 1 , we have For J 2 , applying the commutator estimate Lemma 2.6 yields where 1 n + 1 q = 1 2 . For the first term on the right-hand side of (4.5), by the embedding theorem [13], x , s > n 2 ; and by the L 2 -L q estimate Corollary 3.4, For the second term on the right-hand side of (4.5), by Lemma 3.7, we have Consequently, For J 3 , note that |∇ v w| w, integration by parts yields where we applied the assumption s > n 2 in the last line. For J 4 , we have . Plugging all the estimates from J 1 to J 4 into (4.3), we obtain, Integrating over [0, t] on both sides of of (4.11), we deduce, (E(f k+1 (τ ))).
Inductively, assume (4.12) Taking T 0 sufficiently small such that i.e., (4.14) Inductively, i.e., we get a uniform-in-k estimate. As a routine, let k → ∞, we obtain the solution and complete the proof of existence.
Let us move on to proving the uniqueness. Proof of uniqueness. In the second part, we apply a similar trick in the proof of existence.
Proof. Assume another solution g exists such that taking the difference of f and g, we have (4.17) Applying D s x to (4.17) 1 , we have Multiplying (D s x (f − g)) · w on both sides of (4.18), and then integrating over R n x × R n v yields that 1 2 For J 10 , which is similar to J 4 , we obtain (4.24) x,v (w) . Collecting all the estimates from J 5 through J 10 , and plugging into (4.19), we have, x,v (w) · f − g 2H s . (4.25) Integrating over [0, t] on both sides of (4.25), we get x,v (w) · f − g 2H s dτ. Recall x,v (w) dτ  By Gronwall's inequality, we have f − g 2H s = 0, i.e., f ≡ g, which completes the proof of uniqueness. Thus, we end the proof of Theorem 2.7.