WELL-POSEDNESS FOR THE KELLER-SEGEL EQUATION WITH FRACTIONAL LAPLACIAN AND THE THEORY OF PROPAGATION OF CHAOS

This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term −ν(−∆) α 2 ρ (1 < α < 2). Firstly, the global existence of weak solutions is proved for the initial density ρ0 ∈ L1∩L d α (Rd) (d ≥ 2) with ‖ρ0‖ d α < K, where K is a universal constant only depending on d, α, ν. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in Lr for any 1 < r < ∞. Secondly, for the more general initial data ρ0 ∈ L1 ∩ L2(Rd) (d = 2, 3), the local existence is obtained. Thirdly, for ρ0 ∈ L1 ( Rd, (1 + |x|)dx ) ∩ L∞(Rd)( d ≥ 2) with ‖ρ0‖ d α < K, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant α-stable Lévy process Lα(t). Also, we prove the weak solution is L∞ bounded uniformly in time. Lastly, we consider the N -particle interacting system with the Lévy process Lα(t) and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment ∫ Rd |x| ρ0dx for some 1 < γ < α is below a universal constant Kγ and ν is also below a universal constant. Meanwhile, we prove the propagation of chaos as N →∞ for the interacting particle system with a cut-off parameter ε ∼ (lnN)− 1 d , and show that the mean field limit equation is exactly the generalized KS equation. 2010 Mathematics Subject Classification. Primary: 65M75, 35K55; Secondary: 60J70.

Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in L r for any 1 < r < ∞.Secondly, for the more general initial data ρ 0 ∈ L 1 ∩ L 2 (R d ) (d = 2, 3), the local existence is obtained.Thirdly, for with ρ 0 d α < K, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant α-stable Lévy process Lα(t).Also, we prove the weak solution is L ∞ bounded uniformly in time.Lastly, we consider the N -particle interacting system with the Lévy process Lα(t) and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment R d |x| γ ρ 0 dx for some 1 < γ < α is below a universal constant Kγ and ν is also below a universal constant.Meanwhile, we prove the propagation of chaos as N → ∞ for the interacting particle system with a cut-off parameter ε ∼ (ln N ) − 1 d , and show that the mean field limit equation is exactly the generalized KS equation.

HUI HUANG AND JIAN-GUO LIU
1. Introduction.In this paper, we study the existence, uniqueness, stability and regularity for the following generalized Keller-Segel model with nonlocal diffusion term −ν(−∆) where ν is a positive constant.As usual, this model is developed to describe the biological phenomenon chemotaxis with anomalous diffusion.In the context of biological aggregation, ρ(t, x) represents the density of some biology cells, c(t, x) represents the chemical substance concentration and it is given by the fundamental solution where , i.e. α d is the volume of the d-dimensional unit ball.The motivation of using anomalous diffusion comes from the fact that in many situations found in nature, organisms adopt Lévy process search strategies which have continuous paths interspersed with random jumps (also called Lévy flight) and therefore dispersal is better modelled by the non-local operator such as −(−∆) α 2 [2,19,20,25,26].Indeed, experimental evidences of super-diffusive behaviour have been found in biological systems, such as microzooplankton [25], soil amebas [26] and E. coli [32].Super-diffusion is characterized by a super-linear dependence in time of the mean square-displacement of the position of the dispersing population.In mathematical description: the normal diffusion's variation satisfies [X(t), X(t)] ∝ t, and the super-diffusion's variation satisfies [X(t), X(t)] ∝ t β with β > 1.Moreover, the correct description of a population undergoing super-diffusion is Lévy process.
As the simplest Lévy process, the rotationally invariant α-stable Lévy process has the infinitesimal generator of the form (−∆) α 2 , 0 < α < 2 [1,3], see also [38].For readers' convenience, we give a brief introduction to the α-stable Lévy process in Appendix A. In probabilistic terms, by replacing the Laplacian to its fractional power, we can extend the results for the stochastic equations driven by Brownian motion to those driven by α-stable Lévy process.In [31], the uniqueness and stability under Wasserstein metric of classic KS equation have been proved by associating it with a self-consistent stochastic process driven by Brownian motion.This provides us a similar method to prove the uniqueness and stability for nonlocal KS equation (1) by considering a self-consistent stochastic process driven by rotationally invariant α-stable Lévy process (see Section 4).
Compare system (1) with the classic model of chemotaxis introduced by Keller and Segel in [24].The difference is that we replace the Laplacian ∆ by its fractional power −(−∆) α 2 which is a integral operator, namely the fractional diffusion with exponent 0 < α < 2. In recent years, there has been a surge of activity focused on the use of this fractional diffusion operator, such as [13,14,15] by Caffarelli et al..The main reason for using fractional Laplacian is that we can further extend the theory of diffusion by taking into account the presence of the so-called long range interactions.This nonlocal operator does not act by point-wise differentiation but by a global integration with respect to a singular kernel.We refer to [39,40] for comprehensive review of recent progress in the theory of fractional Laplacian operator.
Under the mass invariant scaling ρ λ (t, x) = λ d ρ(λx, λ α t), KS model (1) exhibits the supercritical behavior.Namely, the aggregation dominates the diffusion for high density (large λ) and the density may blow up in finite time.While for low density (small λ), the diffusion dominates the aggregation and the density has infinite-time spreading.Also, notice that PDEs (1) possesses L q norm invariant with q := d α .Indeed if ρ(t, x) is a solution then ρ λ (t, x) = λ α ρ(λx, λ α t) is also a solution, and this scaling preserves the L q norm ρ λ q = ρ q .This invariant scaling will provide us a sharp initial condition ρ 0 q < K in the proof of global existence (see Section 2).
The fractional KS system was first studied by Escudero in [20], where the author prove that this model has blowing-up solutions for large initial conditions in dimensions d ≥ 2. Also, he obtains the global existence with the initial data , which is a subcritical case.This system has also been studied by Biler et al. [6,7,8,9,10] and Li et al. [27,28,29].For example, in [7], authors study the conditions for local and global in time existence of positive weak solutions in dimensions d = 2, 3.In [8], authors deal with the socalled mild solutions based on applications of the linear analytic semigroup theory to quasi-linear evolution equations.They prove the existence of local in time mild solutions and global mild solutions under the small initial data ρ 0 q < ε in dimensions d ≥ 2. In [6], authors consider the Keller-Segel model for the chemotaxis with either classical or fractional diffusion in dimension d = 2.The blow-up of solutions in terms of suitable Morrey spaces norms is derived.In [29], the authors prove the local existence and uniqueness of solutions by assuming ρ 0 ∈ L p ∪ H s (R 2 ) with s > 3 and 1 < p < 2.Moreover, they attain further properties of the solutions including mass conservation and non-negativity.
Compared to the former studies, ours has more evolved results: I. (Global existence, hyper-contractive and decay estimates) where K is a universal constant only depending on d, α, ν.We prove that there exists a global weak solution ρ such that ρ(t, •) d α < K for all t > 0. The mass conservation, decay estimate and hyper-contractivity are also obtained (see Theorem 2.3).

II. (Local existence)
We have proved that under the more general initial density 0 Before we go to further results, we recast c in (2) as c = Φ * ρ where Φ(x) is the Newtonian potential, and it can be represented as ( Thus we have the attractive force where C * = Γ(d/2) 2π 2/d .Moreover ∇c = F * ρ.In this paper, we introduce the following mean-field self-consistent stochastic process X(t) underlying the KS equations: where X 0 has density ρ 0 (x), and L α (t) is a rotationally invariant α-stable Lévy process.Furthermore, we require the process X(t) has the density ρ(t, x) and the drift term R d F (x − y)ρ(s, y)dy is self-determined.Next we introduce the following notion of strong solution of (5) by requiring ρ ∈ L ∞ 0, T ; L 1 ∩ L ∞ (R d ) for any T > 0 to make sure the log-Lipschitz continuity of the self-consistent term R d F (x − y)ρ(s, y)dy.This kind of log-Lipschitz continuity also appeared in the 2D incompressible Euler equation and the uniqueness was proved by Yudovich [42].
Definition 1.1.We say that X(t), ρ is a strong solution to (5) if there is a stochastic process X(t) and it has the density ρ We will utilize the strong solution of ( 5) as a characteristic line to prove the uniqueness and stability for the KS equation (1) under the following assumptions: Assumption 1.For 1 < α < 2, the initial data ρ 0 (x) satisfies: III. (Uniqueness) For d ≥ 2, the initial data ρ 0 satisfies Assumption 1.We obtain a unique global weak solution ρ(t, x) to (1) with regularity , for any T > 0.Moreover, the corresponding self-consistent stochastic equation ( 5) has a unique strong solution X(t), ρ with initial data (X 0 , ρ 0 ), and ρ is the unique weak solution to (1) (see Theorem 4.2).IV. (Dobrushin's type Stability) With the help of self-consistent stochastic process (5), we also obtain the stability with initial data in Wasserstein distance W 1 for (1).Namely, for any fixed T > 0, there exists two constants C depending on and C T depending only on T such that for any t ∈ [0, T ] sup , where ρ 1 t , ρ 2 t are weak solutions to (1) with initial data ρ 1 0 (x), ρ 2 0 (x) respectively (see Theorem 4.3).
Our last result will deal with the following N -particle interacting system of many indistinguishable individuals {X i (t)} N i=1 with Newtonian potential aggregation and N independent rotationally invariant α-stable Lévy process {L i α (t)} N i=1 : under the condition that the initial data {X i 0 } N i=1 are independent and identically distributed (i.i.d.) with a common probability density function ρ 0 (x).
Although we can only prove the collision happens when R d |x| γ ρ 0 (x)dx is below a certain constant, we believe that the collision for (1) is generic since the initial data ρ 0 may concentrate in a local region.Therefore in order to obtain a global strong solution to the interacting particle system, we regularize the force F (x) by a blob function J(x) ∈ C 2 (R d ), supp J(x) ⊂ B(0, 1), J(x) ≥ 0 and B(0,1) J(x)dx = 1.
In this article we take a cut-off function J(x) ≥ 0, J(x) ∈ C 3 0 (R d ), where C is a constant such that The regularized particle system is given by This system has a unique global strong solution {X i ε (t)} N i=1 by a standard theorem for stochastic differential equations (SDEs) [33,pp.249,Theorem 6].VI. (Propagation of chaos) Assume the initial density ρ 0 satisfies Assumption 1.
Let {X i (t)} N i=1 be the unique strong solution to (5).We prove the propagation of chaos for the interacting system with a cutoff parameter ε ∼ (ln N ) − 1 d , i.e.
(see Theorem 5.5).Noticing that in the case α = 2, the generalized KS equation (1) reduces to the classic KS equation and the stable Lévy process L α (t) reduces to the Brownian motion which has been studied in [31].In the following sections, our discussion will focus on the case 1 < α < 2, but the same results for α = 2 can be obtained similarly.
Concluding this introduction, we present the outline of the paper.
In Section 2, we start with the definition of fractional Laplacian and its basic properties.As a preliminaries, some useful functional inequalities are introduced too.The main results in this section are the global existence and hyper-contractive estimates.Then, the local existence is given in Section 3. Section 4 is devoted to the well-posedness for the generalized KS equation and its corresponding self-consistent stochastic equation.In Section 5.1, we show that the expectation of the collision time for the particle systems is bounded by a universal constant, and then we prove the propagation of chaos in Section 5.2.In the Appendix A, we introduce the definition and basic properties of the rotationally invariant α-stable Lévy process, and the proof of L ∞ uniform bound is given in Appendix B.

Global existence with initial data
2.1.Preliminaries.According to Stein, Chapter V in [35], the definition of the nonlocal operator (−∆) α 2 , known as the Laplacian of order α 2 , is given by means of the Fourier multiplier where ρ(ξ) = F ρ(x) is the Fourier transformation of ρ(x).
Also, we will use the following formula as in [13], which is useful to study local properties of equations involving the fractional Laplacian operator where is a normalization constant and P.V. denotes the Cauchy principle value.Then observe that the following properties hold: Next, we give the definition of weak solution to the KS equation (1).
Definition 2.1.Assume the initial data 0 ≤ ρ 0 (x) ∈ L 1 (R d ), and T > 0. We say ρ(t, x) is a weak solution to (1) with initial data ρ 0 (x) if it satisfies 1. Regularity: 3. c is the chemical substance concentration associated with ρ and given by Remark 1.Notice that the regularity ( 8) is enough to make sense of each term in (10).By the Hardy-Littelwood-Sobolev inequality one has Now, We recall here some useful inequalities which will be used throughout the paper: (Stroock-Varopoulos' inequality) [12] Let 0 where the best constant is given by .
(ii) For c = 1, y(t) uniformly bounded Notation.Without confusion, we denote the L p norm of a function by f p .Also, we will use the Sobolev space of non-integer power W s,p (R d ), s ∈ R, which is defined via the Fourier transform F: Specifically, when p = 2, we have H s (R d ) with the norm defined as Inessential constants will be denoted generically by C, even if it is different from line to line.

2.2.
Global existence and hyper-contractivity.In this subsection, we derive the global existence of weak solutions in a standard approach.Firstly, we define two constants which are related to the initial condition for the existence results: ) and ζ > 0, then there exists a global weak solution ρ such that ρ(t, x) q < K for all t > 0. Furthermore, (i) For any T > 0, we have the following regularity (ii) The weak solution satisfies mass conservation and the following hyper contractive estimates hold true for any t > 0 and any 1 < r < ∞: where 0 satisfies Proof.The proof can be divided into 9 steps.Steps 1-6 give some crucial priori estimates for the statement (i), (ii).In Steps 7-9, a regularized equation is constructed to make these priori estimates of Steps 1-6 rigorous and obtain the global existence of a weak solution to (1).
For the rigorous proof, we follow the method in [4] by taking a cutoff function 0 and D α ψ 1 (x ) is finite for x ∈ R d .This cutoff function will be used to derive the existence of the weak solution.
Step 1. (Uniform L q estimates) Firstly, it is obtained by multiplying (1) with qρ q−1 and using (13), then integrate over R d Compute the right side and use ( 14) which implies Since ρ 0 q < K, so the following estimates hold true and ζ = K − ρ 0 q , from the equations above one has which leads to the following estimates Step 2. (L q decay estimates) By using ρ 1 ≤ ρ 0 1 compute which leads to which leads to the decay property ) −q(q−1) ρ 0 q 1 t −q(q−1) .
Step 3. (Uniform L r0 estimates with r 0 = q + 0 for 0 small enough) As we have done before If we choose 0 such that then one has On the other hand where θ = r0(q−1) q(r0−1) , and it leads to

1
, and here δ = 1 + 1 q(r0−1) , which implies r 0 q(r 0 −1) 1 , then one computes Step 4. (Hyper-contractive estimates of L r norm for r > r 0 ) For r > r 0 we compute as before by using the Young's inequality where θ = qr[r0−(r+1)] (r+1)[r0(q−1)−qr] satisfying 2θ(r+1) r < 1 for r 0 > q.Collecting (23) yields Thus we have for any t > 0 with 0 satisfying (20) Step 5. (Decay estimates on ρ r ) In this step, based on the decay of ρ q with time evolution, ρ r decays for large time.Divide r into two cases 1 < r < q and q < r < ∞.Recalling that in Step 2 we have gotten (1) For 1 < r < q, it follows from ( 27) by applying the interpolation inequality that for any t > 0, (2) For q < r < ∞, since ρ q decays to zero as time goes to infinity, then for t larger than some T r one has Solving this ordinary differential inequality, the large time decay of ρ r has been obtained Step 6. (Mass conservation) Observe that for any t > 0 Using the interpolation inequality, we have Thus as R → ∞ by the dominated convergence theorem one has Combing the virtue of ( 26), ( 28) and ( 29), the statement (ii) has been proved.
Step 7. (Regularization) In order to show the existence of a weak solution with the above prosperities and make the proof rigorous, we consider the following regularized problem for ε > 0: ) is defined by J(x) as in (6).From parabolic theory, the regularized problem has a global smooth positive solution ρ ε with the regularity ρ ε (t, x) r ≤ C ε for all r ≥ 1, t > 0. By taking the similar arguments as in Step 6 arrives at the mass conservation of ρ ε .
Multiply equation (31) with rρ r−1 ε ψ R (x), then integrate over R d , one has By using the Hardy-Littlewood-Sobolev inequality, we know .
≤ C ε , then the last two terms of (32) will vanish as R → ∞.Thus the following inequality holds Therefore all the estimates in Steps 1-5 hold true.
For the initial density 0 ), the following basic estimates are obtained: In addition, for any T > 0, applying the weak Young's inequality, one has Step 8. (Time regularity and application of Aubin-Lions-Dubinskiȋ lemma) In order to get the regularity of ∂ t ρ ε , one takes any test function and estimate ∂ t ρ ε , h .We have Thus for any T > 0 Finally the regularity of ∂ t ρ ε follows Before we use the Aubin-Lions-Dubinskiȋ lemma, we introduce the so called seminormed nonnegative cone M + in Banach space B: M + ∈ B; for all u ∈ M + and c ≥ 0, cu ∈ M + ; there exist a function [•] : M + → [0, +∞) such that [u] = 0 if and only if u = 0, and [cu] = c[u] for all c ≥ 0.
For any bounded domain Ω, we choose B = L q (Ω), and define M + (Ω) := {ρ : It is easy to check that M + (Ω) defined here is a seminormed nonnegative cone in L q (Ω).
For q ≥ 2, one has For 1 < q < 2, we set u ε = ρ q 2 ε and u = ρ q 2 , by the mean value theorem and Hölder inequality, one has Thus, we get Recall that . By Aubin-Lions-Dubinskiȋ lemma as in [17], one arrives at that {ρ ε } ε>0 is compact in L q 0, T ; L q (Ω) .Consequently, there exists a subsequence ρ ε without relabeling such that k=1 ∈ R d be a sequence of balls centered at 0 with radius R k , R k → ∞.By a standard diagonal argument, there exists a subsequence ρ ε without relabeling the following uniform strong convergence holds true Step 9. (Existence of a global weak solution) Now, we will prove that ρ is a weak solution to (1).Indeed, the weak formulation for ρ ε is that for any ϕ ∈ C ∞ c (R d ) and any 0 < t < ∞, For the first term of the right side of (41), it is obvious that ]dxds, as ε → 0.
(42) For the second term, since F ε (x) = F (x) for any |x| ≥ ε , one has Firstly, by using the fact For I 11 , by using the Hardy-Littlewood-Sobolev inequality, one has By using regularity (34) of ρ ε , from (45), we know Similarly, we can obtain t 0 I 12 ds → 0 as ε → 0, which leads to Same as the discussion of I 11 , (48) leads to t 0 I 21 ds → 0, as ε → 0. And similarly, we have t 0 I 23 ds → 0. As for I 22 , we have From ( 49), one has where we have used the regularity (34) for ρ ε and ρ.Hence we have From all the discussion above, one obtain (51) Combining ( 42)and (51), we conclude ρ is a weak solution to (1).By now we have given the existence of a global weak solution, and the conservation of mass is easy to attain by the similar arguments as in Step 6.
Then there exists T > 0, such that a weak solution to (1) ρ(t, x) exits in [0, T ] with regularity and the conservation of mass Proof.The result can be found in [10], but for the completeness, we will give the sketch of the proof here.
Multiply equation (31) with 2ρ ε ψ R (x), then integrate over R d , one has By using the Hardy-Littlewood-Sobolev inequality, we know .
As we have done in the last section, the last two terms of (32) will vanish as R → ∞.Thus the following inequality holds For α ≥ d 3 , we can use the interpolation inequality and the Sobolev imbedding theorem (H s → L For α > d 2 , we imply the Young's inequality Thus we have Solving above ordinary differential inequality, we obtain which implies there exists a T ( ρ 0 2 2 ) independent of ε such that for t ∈ [0, T ], the following estimates hold Now we can use the Lions-Aubin lemma, there exists a subsequence ρ ε without relabeling such that for any ball B R , and ρ(t, x) is a weak solution to (1).The regularity of ρ follows: Moreover the conservation of mass can be proved as we have done in the Step 6 from last section.
for some r > d α := q, then there are T > 0 and a weak solution ρ(t, x) in 0 < t < T to (1) with mass conservation.
Proof.As in (16), it yields And notice that where ∆P A (s) = P A (s) − P A (s−) is the jump increment.Since Lévy process has independent increments, we know X ε (s−) and ∆P A (s) are independent.Moreover, it follows from [1, Theorem 2.3.7] that Hence we have We take A = {x : |x| ≥ 1} in (65), thus Denote A δ = {x : 0 < δ ≤ |x| < 1}, and Moreover, we can prove that g δ is a Cauchy sequence.Actually, for 0 < δ 1 < δ 2 , we have |f (x)|µ (dx)ds, (68) by (64).It follows from ( 67) and (68) that Now we apply ( 65) and ( 69) where we have used the Dominated convergence theorem in the last equality.Now combing (66) and (70), we can take expectation on both side of (62) , then one has |x| d+α dx in (71), and it leads to Then we use the properties of fractional Laplacian in (7), one has which leads to Thus we know ρε (t, x) satisfies the following equation in distribution sense Since ρ ε (t, x) is also a weak solution to (74) and the weak solution of (74) is unique, then we get ρε (t, x) = ρ ε (t, x).It means that X ε (t), ρ ε (t, x) is a strong solution to (58).The uniqueness of the strong solutions to (58) comes from the uniqueness of the solutions to (60).In fact, suppose X 1 ε (t), ρ 1 ε and X 1 ε (t), ρ 2 ε are two solutions to (58).By the Itô formula we have used before, one knows the ρ 1 ε and ρ 2 ε both are weak solutions to (60) with the same initial data ρ 0 (x).Since the weak solution to (60) is unique, one has

Existence, uniqueness and stability with initial data
) and ρ 0 q < K.The uniqueness of weak solutions to the KS model has been concerned by many scholars.The optimal transport method [16] and the renormalizing argument [18] have been used to prove the uniqueness of weak solutions to the classical KS model with normal Laplacian term.Here we will follow the method in [31] to prove the uniqueness for the generalized KS model ( 1).Now we introduce a topology of the Wasserstein space which is useful in proving the following theorem.Consider the space of probability measure We define the Kantorovich-Rubinstein distance in P 1 (R d ) as follows where Λ(f, g) is the set of joint probability measures on R d × R d with marginals f and g.And it has been proved that P 1 (R d ) endowed with this distance is a complete metric space in [41,Theorem 6.18].Also we will use the following time dependent measure space L ∞ 0, T ; Moreover, it is a complete metric space equipped with metric for any two elements Theorem 4.2.Assume the initial density ρ 0 satisfies Assumption 1, then for any T > 0 and t ∈ [0, T ], (i) There exists a unique weak solution ρ(t, x) to (1) with initial density ρ 0 and regularity (ii) There exists a unique strong solution X(t), ρ to (5) with initial data (X 0 , ρ 0 ), and ρ is the unique weak solution to (1).
Proof.The sketch of the proof will be divided into 4 steps and we refer to [31, Theorem 1.1] for more details.
Step 1. (Global existence and L ∞ (R d ) uniform bound ) The global existence of weak solution to (1) has been proved in Theorem 2.3.Following the method in [30], we leave the proof of the uniform L ∞ estimate in Appendix B.
Step 2. (Existence of strong solution to ( 5)) Firstly, we give some uniform estimates for the regularized equation.For ε > ε > 0, consider equation (58), and suppose X ε (t), ρ ε (t, x) , X ε (t), ρ ε (t, x) are two strong solutions in Theorem 4.1 starting from the same initial data X 0 .One can show that there exists a constant C T and ε 0 (T ) such that if ε < ε 0 (T ), then where Consequently, there exists a stochastic process On the other hand there exists a unique since L ∞ 0, T ; P 1 (R d ) is a complete metric space.Secondly, one can show the limiting density is the weak solution to KS equations (1), i.e.
where ρ(t, x) is a weak solution of (1).
Lastly, we conclude that the limiting stochastic process X(t) is the strong solution to (5).
Step 3. (Uniqueness of strong solutions to (5)) Assume X(t), ρ(t, x) , X (t), ρ (t, x) , are strong solutions to (5) with the same initial data.Then one can deduce that Here ω(x) is defined as which is related to log-Lipschitz continuity of the field F (x − y)ρ(s, y)dy, seen in [31,Lemma 2.2].
Step 4. (Uniqueness of weak solutions to (1)) Suppose ρ , ρ are two weak solutions with the same initial density ρ 0 .For any fixed random variable X 0 with density ρ 0 , by the following Proposition, there exists two processes X(t) and X (t) such that X(t), ρ(t, x) , X (t), ρ (t, x) both are strong solution to (5) with the same initial data X 0 , ρ 0 .Therefore (81) holds, the uniqueness is proved.
Proposition 1.The relation between weak solution to (1) and strong solution to (5) can be described (i) If X(t), ρ is a strong solution to (5) with initial data X 0 , ρ 0 , then ρ(t, x) is a weak solution to (1) with initial data ρ 0 .
(ii) If ρ(t, x) is a weak solution to (1) with initial data ρ 0 (x), then for any X 0 with density ρ 0 (x), there is a unique process X(t) with density ρ(t, x) and X(t), ρ is a strong solution to (5) with initial data X 0 , ρ 0 .
Proof.The proof of this proposition is similar to [31,Proposition 2.3] except that we use a different Itô formula as we have done in Theorem 4.1.
Furthermore, with the help of the self-consistent stochastic process of ( 5), we also obtain the following stability with initial data in the Wasserstein distance for (1).
)dx be two weak solutions to (1) with initial data ρ 1 0 (x), ρ 2 0 (x) respectively and they satisfy Assumption 1. Then there exists two constants C depending on and C T depending only on T such that where W 1 is the Wasserstein distance.
Proof.The proof of this theorem is similar to [31, Theorem 1.2] except that we change Brownian motion into rotationally invariant α-stable Lévy process..

5.
Interacting particle system and mean-field limit.Inspired by [31], we introduce the stochastic system of interacting particles with singular force kernel and rotationally invariant α-stable Lévy process described as follows.Let (Ω, F, P) be a probability space equipped with a filtration (F t , t ≥ 0) which satisfies the usual hypothesis of right continuity and completion, i.e.F is complete and F t is right continuous.We suppose that the space is endowed with N independent ddimensional rotationally invariant α-stable Lévy process {L i α (t)} N i=1 .Furthermore, every Lévy process L i α (t) will be assumed to be F t -adapted which have càdlàg (right continuous with left limits) simple paths, and L i α (t) − L i α (s) is independent of F s for all 0 ≤ s < t < ∞.And with the assumption α ∈ (1, 2], it allows us to freely use expectation of the α-stable process.Denote {X i (t)} N i=1 be the positions of Nparticles at time t, where X i (t) ∈ R d .The initial data {X i 0 } N i=1 are the i.i.d.random variables with a common probability density function ρ 0 (x).Moreover we assume the particles in the system interact with each other by Newtonian potential Φ(x) as in (3), and we have the interacting force F (x) = ∇Φ(x).Thus the dynamics of the interacting particle system can be described by a system of stochastic differential equations Various interesting particle systems in physical and biological science can described by this equation.We can see the first term in the right hand side of (82) represents the attractive force on X i (t) by all particles.Moreover, we assume the initial data {X i 0 } N i=1 are i.i.d.random variables with the same distribution L(X i 0 ) = f 0 (x) and density ρ 0 (x).
For (82), particles may collide to each other due to attractive force.Hence we consider a standard smoothing kernel J ε (x) satisfying J ε (x) = 1 ε d J( x ε ), where J(x) is defined as in (6).Let F ε = J ε * F , then regularized system has a unique strong solution {X i ε (t)} N i=1 and F ε (x) = F (x) for any |x| > ε [31, Lemma 2.1].

Collision between particles.
In this subsection, we show that the expectation of the collision time for the interacting particle system (82) is below a constant.
The following lemma is a useful result from the process of proving Theorem 2.3 in [8].
and let τ = lim ε→0 τ ε .There exist universals two constants K γ , T c > 0, such that if Proof.Adapting the method of proof of in [31,Theorem 3.1], we know the system (82) has a unique strong solution until the explosion time τ = sup{t ∧ T : inf is the global unique solution to the regularized interacting system (83).By Itô formula, we choose where we have used Lemma 5.1.Sum all of (86), we get Since X i ε (t) is the unique solution to (82) and where in the second inequality we have used Lemma 5.2.Take expectation of (87) and by exchangeability of X i ε (t), one has From Lemma 5.3, For certain (sufficiently small) universal constant If we choose 0 < K γ < 1 2 ( K2 νK1 ) By the positivity of the left handside of (90), one has Finally, by the monotone convergence theorem, we concludes the proof.

Propagation of chaos.
The concept of the propagation of chaos was originated by Kac [23].It is important for the kinetic theory that serves to relate the kinetic equations, such as the Fokker-Planck, Boltzmann and Vlasov equations.In this subsection we prove the propagation of chaos for the KS equations ( 1) following the method in [31].We refer to [11,22,36,37] for more instances of the propagation of chaos.
Proof.We will only give a sketch of proof here since it is similar to [31,Theorem 1.3].The main idea is to link (83) with ( 5) through (58).In Theorem 4.1, we stated the existence and uniqueness for strong solutions to (58), which derives that if initial data {X i 0 } N i=1 are i.i.d. and Lévy process {L i α (t)} N i=1 are independent, then the following nonlinear stochastic differential equations is the unique strong solution to (83) with the same initial data {X i 0 } N i=1 and Lévy process {L i α (t)} N i=1 .Then for any ε > 0, 1 ≤ i ≤ N and T > 0, we can prove where C T is a constant independent of ε.The detail of the proof to (94) can be find in [31, Proposition 3.1].On the other hand, similar to (76), there exists a constant C T and ε 0 (T ) > 0 such that if ε < ε 0 (T ) for any ε > 0, 1 ≤ i ≤ N and T > 0, one has Combine ( 94) and (95) together, one has iii) X is stochastically continuous, i.e. for all a > 0 and for all s ≥ 0 lim such that, for each t ≥ 0, An important by-product of the Lévy-Itô decomposition is the Lévy-Khintchine formula: If X is a Lévy process then for each u ∈ R d , t ≥ 0, where B = {y ∈ R d , |y| < 1}, µ is the Lévy measure.The triple (b, A, µ ) is called the characteristics of the Lévy process.For a Lévy process X(t) we have where η X(1) is the Lévy symbol of X(1), which can be seen in [1,Therorem 1.3.3].When we say the Lévy symbol of a Lévy process X(t), it means the Lévy symbol of the random variable X(1).
A Lévy process X(t) is called stable if in which each X(t) is a stable random variable.Of particular interest is the so called rotationally invariant α-stable Lévy process, where the Lévy symbol is defined by (100), i.e. η X(1) (u) = −σ α |u| α (0 < α < 2).For simplicity we choose σ = 1 in the sequel.And we denote this particular rotationally invariant α-stable Lévy process as L α (t), and the characteristics of it is (0, 0, µ ), where [1, pp.37].Specifically, we choose C to be C d,α as in the definition of fractional Laplacian operator.Moreover we have Appendix B. Uniform L ∞ estimate.First, we will give a proof for the L r (R d ) (q < r < +∞) bound uniformly in time in the following lemma.

)
To better understand the Poisson stochastic integrals [1, P.231], let A be an arbitrary Borel set in R d − {0} which satisfies inf x∈A |x| ≥ C > 0, and denote P A

t→sP
(|X(t) − X(s)| > a) = 0. Then we have to mention the famous Lévy-Itô decomposition: If X(t) is a Lévy process, then there exists b ∈ R d , a Brownian motion B A with covariance matrix A, an independent Poisson Random measure N on R + × (R d − {0}) and corresponding compensator Ñ = N − E(N ),