A GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE TWO-DIMENSIONAL VLASOV-FOKKER-PLANCK AND MAGNETOHYDRODYNAMICS EQUATIONS WITH LARGE INITIAL DATA

. We present a two-dimensional coupled system for particles and compressible conducting ﬂuid in an electromagnetic ﬁeld interactions, which the kinetic Vlasov-Fokker-Planck model for particle part and the isentropic compressible MHD equations for the ﬂuid part, respectively, and these separate systems are coupled with the drag force. For this speciﬁc coupled system, a suﬃcient framework for the global existence of classical solutions with large initial data which may contain vacuum is established.

The purpose of this paper is to provide a global existence theory of classical solutions to the coupled compressible magnetodrodynamic-Vlasov-Fokker-Planck system describing the interaction of the compressible conducting fluid in an electromagnetic field and particles. This kind of coupled system arises out of many practical industrial applications such as the biosprays in medicine [2], sedimentation of solid grain in physics [4,6], fuel-droplets in combustion theory [36] etc.
For notational simplicity, we denote that the operator Γ ρ u is defined by where the shear, bulk viscosity and magnetic diffusive coefficients µ, λ and ν are assumed to satisfy such that Γ ρ is strictly elliptic.
Moreover, ρ f and u f denote the local mass density, and average local velocity of particle ensemble, respectively: f dv and u f := The modeling of collective dynamics via a coupled kinetic-fluid system is one of the hottest topics in the field of nonlinear partial differential equations in recent years. When the number of flocking particles is sufficiently large, it is almost impossible to track the motion of each particle. Therefore, we use the corresponding VFP type kinetic equation to describe the motion of particles [35]. On the other hand, the fluid dynamics in an electromagnetic field can be described hydrodynamic models such as compressible MHD equation, please refer to [21,22] for more details. Next, we briefly review some earlier results on coupled system. There are many literatures on the coupled system between particles and the compressible flow. In [32,33], a global weak solution is constructed and the asymptotic analysis has been studied for the coupled system of the VFP equation with the compressible Navier-Stokes (NS) equations in a bounded domain Ω ∈ R 3 . In [13], the authors proved the global existence of classical solutions to the coupled system of VFP equation and compressible Euler equations for small initial data in the whole space R 3 . In [27], the global well-posedness of a strong solution to compressible Navier-Stokes-Vlasov-Fokker-Planck system in the three-dimensional whole space is established when the initial data is a small perturbation of some given equilibrium. The global classical solutions to 1D and 2D coupled system of flocking particles and compressible fluids with large initial data have been obtained in [18,19]. Furthermore, we can refer to [1,3,5,6,20,17] for more information on the coupled system. Especially, when the influence of distribution function f (t, x, v) is neglected, the system (1) becomes isentropic compressible MHD equations. This kind of Vaigant-Kazhikhov model for compressible NS equations is considered firstly in [37]. Later on, the global existence and uniqueness of classical solutions to Vaigant-Kazhikhov model of compressible NS equations and compressible MHD equations are established on torus T 2 and the whole space R 2 in [23,24,25,26,30]. Specifically, Caffarelli-Kohn-Nirenberg inequality is a essential tool for dealing with this system in the whole space R 2 .
Up to now, a natural question is raising, can we get the global solvablity of coupled system (1) with large initial data? This paper gives a positive answer, even to the initial density ρ(t, x) contains vacuum. We briefly state the main ideas and techniques in the process of proof. Firstly, the friction term in (1) 3 has highly nonlinearity. It requires us deal with the fluid velocity u(t, x) and kinetic distribution function f (t, x, v) in the mean time. The estimate I 11 in Lemma 2.2 gives the basic skill, and the high order estimates of friction term have analogous results. Moreover, L p x estimates of the momentum for kinetic part f (t, x, v) in Corollary 9 have been used in several times. Secondly, according to the blow up criterion of the compressible flow in [24], in order to get the existence and continuation in time of flows, the L ∞ x norm in space variable of density ρ(t, x) need to be bounded. Based on the above fact, we should control the upper bound of density ρ(t, x). The upper bound of density ρ(t, x) is obtained in Lemma 2.8, with the help of introducing the nonlinear funtionals Z(t), χ(t) in Lemma 2.6 and the Riesz transform. Thirdly, we need to deal with the term of (ρ f ) t , (ρ f ) tt and (u f ρ f ) t in the higher order apriori estimates. Using the equation (1) 1 , we convert these terms into taking space derivative on momentum f (t, x, v) in (63).
The rest of paper is organized as follows. In Section 1, we briefly discuss a framework and present our main results. In Section 2, we provide several lemmas to be used later. In Section 3, we derive apriori higher order estimates of the coupled system. In Section 4, a proof of main result is provided. Notation. Throughout the paper, the definitions of operators are given as fol- . Moreover, C denotes a generic positive constant which may change line by line. The small constants to be chosen are denoted by ε, σ, δ. For function spaces, W k,p (R 2 ) and W k,p (R 2 × T 2 ) denote the standard Sobolev spaces with standard norm · W k,p , and H k := W k,2 . · p := (

2.
Initial data [f 0 , ρ 0 , u 0 , H 0 ] satisfy regularity and integrability: 3. Initial data [f 0 , ρ 0 , u 0 , H 0 ] satisfy the compatibility condition: Then, the Cauchy problem (1)-(2) admits a unique global solution [f, ρ, u, H] satisfying the following regularity and integrability: for any T ∈ (0, ∞), Remark 1. 1. For simplicity, we only require that k is suitably large. Since the weight is used many times throughout the paper, it is really not easy to give a exact expression of k. So we do not track the optimal k. 2. Here, we only consider the friction term dependent on relative velocity. Furthermore, the condition of fluid velocity density ρ(t, x) is contained is worthy of consideration.
3. By introducing the Caffarelli-Kohn-Nirenberg inequality in [26], the coupled system has similar global solvablity in the whole space R 2 .

2.
A priori lower-order estimates. In this section, we present lower-order energy estimates for the coupled system (1), and derive several momentum estimates for f (t, x, v).
Next, we show some momentum (velocity) estimates for the kinetic part f (t, x, v). For this, we set wherek ≥ 2 is a positive constant.
Proof. (i) Note that for R > 0, We now choose R = ( R 2 |v| k2 f dv) 1 k 2 +2 in the above relation to obtain .
• (Estimate of I 11 ): Again we apply the Hölder inequality and the result (i) to obtain .
• (Estimate of I 12 ): We use integration by parts to get In (9), we collect all estimates to find d dt Finally, we integrate the above inequality over [0, t] and use the Gronwall lemma to derive the desired estimate.
We apply the operator div to the momentum equation (1) 3 to have where the effective viscous flux F is defined by On the other hand, consider the following three elliptic problems on the torus T 2 : − ∆ x ψ = div(ρu), For equations (11), we can derive the following elliptic estimates in the following lemma. It can be easily established through a similar way as in [25]. So, we omit it here.
Lemma 2.4. The following estimates hold.
Proof. Now we only prove the last estimate (v), others can be found in [30]. From (iv) in Lemma 3.3, we have On the other hand, we use (ii) in Lemma 2.1, Lemma 2.2 and (ii) in this lemma to have We combine (12) and (13) to derive a desired estimate (v).
It follows from (11) and (10) that which yields We define It follows from the definition of the effective viscous flux F and (1) 2 that Next, we derive the L ∞ t L p x estimate of the density ρ(t, x) by using (14). Lemma 2.5. Let β > 4 3 , and assume that the same conditions in Lemma 2.1 hold.
Proof. We set (h) + to be the positive part of a function h, multiply (14) by ρ[(Λ(ρ)− ψ) + ] 2m−1 with m 1 and integrate the resulting relation over T 2 to obtain 1 2m For the notational simplicity, we define and estimate terms I 2i one by one as follows.
• (Estimate of I 21 ) We use (iv) in Lemma 2.4 to have , and use Lemma 2.4 to have In (16), we collect all estimates to find that We set

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Then, we have

Note that
On the other hand, by (i) in Lemma 2.4 and Young's inequality, we have We use (ii) in Lemma 2.1 and (17) to have We now apply the Gronwall Lemma for (18) Proof. The estimate in (19) For (ii), we only select one term to estimate, the others can be derived similarly.
In fact, we use (i) in Lemma 2.2 for k 1 = 2 and k 2 = 4p − 2 to have Then we use (ii) in Lemma 2.2 for k = 4p − 2, (ii) in Lemma 2.4 and Lemma 2.5 to have

2.2.
Estimates on the fluid density. In this subsection, we provide an upper bound of the fluid density by the method of characteristics. With the help of the Brezis-Wainger inequality, we combine the estimate of ∇u 2 2 and ρu p (p > 3) to derive the upper bound estimate of the fluid density.
We set the material derivative of the fluid velocity byu: and introduce nonlinear functionals: Lemma 2.6. Let T be a positive constant and suppose that the conditions in Lemma 2.1 hold. Then, for t ∈ [0, T ] we have where ε > 0 is a constant which can be arbitrarily small.
Proof. We denote the perpendicular gradient by ∇ ⊥ x := (∂ x2 , −∂ x1 ). Then, the momentum equation can be rewritten as follows: We multiply the above identity byu to obtain These below relations hold: We should handle the strongly coupled magnetic field with the velocity field to obtain the upper bound of the density.
With above, we have On the other hand, we multiply (1) 1 by 1 2 v 2 , integrate over T 2 and use the integration by parts to get and multiply (1) 4 by −C 1 ∆ x H and integrate over T 2 with C 1 > 0 sufficiently large to have We combine (22) − (24) to have (25) Now, we estimate the terms I 3i , i = 1, · · · 13, separately.
• (Estimate of I 31 ): From and C 1 > 0 sufficiently large, we can get • (Estimate of I 32 ): We use the relations the elliptic estimates, Sobolev inequality and Corollary 1 to get Then, the above estimates in (27) yield • (Estimate of I 33 ): By the duality between Hardy H 1 and BMO spaces, we have • (Estimate of I 34 and I 35 ): We use F 2 ≤ Φ for arbitrarily small constant ε > 0. Then we have • (Estimate of I 38 ): From (ii) in Lemma 2.4 and Corollary 1, we have Applying the integration by parts on space variables, we use Corollary 1 to have • (Estimate of I 311 -I 313 ): Collecting the estimates of I 3i (1 ≤ i ≤ 13) in (25), we have Then we further use Lemma 2.5 to derive (20).
The following ρu p with p > 3 will play a crucial role in the estimate of the upper bound of the density as in [23,24].
Proof. Firstly, we derive the estimate of T . To this end, we multiply the momentum equation by (2 + α )|u| α u, integrate over T 2 , and use integration by parts to obtain Now we estimate terms I 4i (1 ≤ i ≤ 4) one by one.

• (Estimate of I 44 ): We use (ii) in Lemma 2.4 and Corollary 1 to have
. Therefore, we combine the above three estimates and (31) to have T , we use interpolation inequality to obtain the result: Now we are in position to prove the upper bound of ρ(t, x).
We integrate the above equation over [0, t] to obtain Now, we estimate the terms on the RHS of (33). For ψ, we use Lemma A.3 to have From the elliptic equation (11) Collecting the estimates (34) and (35), we use (20) to have Now we turn to the estimate of the third term in the RHS of (33). To the end, we first use Lemma 2.1 and (27) to have where in the last inequality one has used Denoting the commutator ϑ = [u i , R i R j ](ρu j ), we use Lemma A.4 and (i) in (30) to have

BINGKANG HUANG AND LAN ZHANG
Then we choose p > 4 sufficiently large and use (37) to have On the other hand, we use the Galiardo-Nirenberg inequality and commutator estimates in Lemma A.4 to get It follows from Lemma 2.5 that Finally, we treat the forth term on the RHS of (33). From the elliptic equation (11) Finally, we substitute all above estimates into (33) to derive When β > 4 3 , we take positive constant ε sufficiently small to have sup 3. A priori high-order estimates. In this section, we present higher-order estimates. Let [f, ρ, u, H] be a classical solution to the coupled system, and we derive some a priori estimates for the system (1)- 3.

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• (Estimate of I 56 and I 57 ): We apply the integration by parts and use (ii) in Lemma 2.4, Corollary 1 and (38) to have Similarly, we have In order to close the (40), we apply operator ∂ t to (1) 4 , multiply the resulting equation by H t and integrate over T 2 to get Similar to the corresponding estimates in Lemma 3.9 [30], we have We collect all estimates of I 6i in (40) and (41) to obtain (26) and (38). We can apply the Gronwall inequality and use (15) to further obtain Proof. We apply the operator ∇ x to the continuity equation (1) 2 , multiply the resulted equation by p|∇ x ρ| p−2 ∇ρ, and integrate over T 2 to get • (Estimate of ∇ 2 x u p ): By the interpolation inequality, we use (39) to have and Then we use (15) 1 , Lemma 2.1 and Corollary 2 to have . On the other hand, we use Lemma 2.1, (15) 1 and Corollary 2 to have 2 + 1). Then we can further estimate ∇ 2 u p as Collecting the estimates of ∇ 2 x u p and ∇ x u ∞ in (44), we have d dt We apply the Gronwall lemma, and use (15) 2 and Lemma 3.1 to have The above inequality together with (45) implies Lemma 3.3. Suppose that the conditions in Lemma 2.1 hold. Then, for 4 < p < +∞ we have Proof. We apply the operator ∂ xi to (1) 1 to obtain We multiply the above equation by v kp p|∂ xi f | p−2 ∂ xi f , and integrate the resulted equations with respect to x, v over T 2 × R 2 to give Now we deal with I 6i , i = 1, · · · , 3, as below.
•(Estimate of I 6i (i = 1, 2)): We use integration by parts to obtain . • (Estimate of I 63 ): Again, we use the integration by parts to have We collect all estimates of I 6i in (47) to find Similarly, we can obtain . Then we combine the above two estimates to have d dt ( We further apply the Gronwall inequality and use (43) to derive (46).
Lemma 3.4. Suppose that the conditions in Lemma 2.1 hold. Then, we have Proof. By the standard L 2 -estimates for (1) 3 , we use Corollary 2, Lemma 3.1 and Lemma 3.2 to have Similarly, we have Combine the above together, we get By the Sobolev inequalities, we use above estimate to have Then we can further use Corollary 2, Lemma 3.1, 3.2to obtain On the other hand, we apply the operator ∇ 2 x to the continuity equation (1) 2 and obtain d dt Similarly, we have
We set m k ∇ x f := R 2 |v| k |∇ x f |dv. Then, by standard elliptic estimates, we use (32), (50) and Lemma 3.2 to have Similarly, (52) We combine (48), (49) and (51) to obtain We apply the Gronwall inequality and use Lemma 3.1, 3.2 to have By the continuity equation (1) 2 , we have Then we use (32) and Lemma 3.2 to have Similarly, we apply the operator ∂ t to (55) to obtain 3.2. Second order estimates. In this subsection, we derive the second order estimates of the classical solution [f, ρ, u, H] to the system (1)-(2): ∇ 2 x (ρ, u) p and v k ∇ 2 x,v f p for 4 < p < +∞.
Lemma 3.5. Suppose that the conditions in Lemma 2.1 hold. Then, for 4 < p < ∞ we have Proof. We apply the operator ∂ t to (1) 2 to obtain We multiply the above equation by u tt , integrate the resulting equation over R 2 and use the integration by parts to obtain

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As the corresponding estimates in Lemma 3.12 [30], the first term, the second term, the fourth term and the last term in the RHS of (60) can be estimated as follows: (61) Now, we turn to treat the third term on the RHS of (60). Rewrite this term as Before the estimation of I 7i , we first note that Applying operator ∂ t with (1) 3 , we have moreover, by the estimates in Lemma 3.4.