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A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data

We present a two-dimensional coupled system for flocking particle-compressible fluid interactions, and study its global solvability for the proposed coupled system. For particle and fluid dynamics, we employ the kinetic Cucker-Smale-Fokker-Planck (CS-FP) model for flocking particle part, and the isentropic compressible Navier-Stokes (N-S) equations for the fluid part, respectively, and these separate systems are coupled through the drag force. For the global solvability of the coupled system, we present a sufficient framework for the global existence of classical solutions with large initial data which can contain vacuum using the weighted energy method. We extend an earlier global solvability result [ 20 ] in the one-dimensional setting to the two-dimensional setting.


1.
Introduction. The purpose of this paper is to provide a global existence theory of classical solutions for the coupled Cucker-Smale-Navier-Stokes system describing the interaction of Cucker-Smale flocking particles and the isentropic viscous compressible fluid. This kind of coupled system arises in many industrial applications such as the sedimentation phenomenon analysis and the modeling of aerosols and sprays. When the flocking particles are surrounded by the viscous fluids, the force per unit mass exerted on a flocking particle by the surrounding fluid will come from several effects, e.g., skin friction, separation drag, gravity and body forces, rotation of the particle with respect to the gas, pressure gradient in the gas, etc (see [37] for details). In this paper, we consider a coupled kinetic-fluid model for the interactions between Cucker-Smale particles and compressible viscous fluid via a friction force in a random environment, which can be modeled by the coupled system of kinetic CS-FP type equation with a degenerate diffusion coefficient and compressible isentropic Navier-Stokes equations. The nonlinear convection term is difficult to be controlled due to the highly nonlinearity of the 3-D Navier-Stokes equations. Therefore, we will consider 2-D case in this paper. To make the coupled kinetic equation be consistent with the fluid equation, we also consider the kinetic equation in 2-D case, which can be rigorously derived by the 2-D microscopic dynamical model as in [22]. More precisely, let f = f (x, v, t) be the one-particle distribution function of a Cucker-Smale (C-S) ensemble with velocity v = (v 1 , v 2 ) ∈ R 2 at position x = (x 1 , x 2 ) ∈ R 2 at time t > 0 for particle side, and let ρ(x, t) be the density and u = u(x, t) = (u 1 , u 2 )(x, t), be the bulk velocity of the compressible fluid. Then the coupled dynamics of [f, ρ, u] is governed by the following kinetic-fluid system: subject to initial conditions: (f (x, v, 0), ρ(x, 0), u(x, 0)) = (f 0 (x, v), ρ 0 (x), u 0 (x)), (x, v) ∈ R 2 × R 2 , (2) where P (ρ) is the pressure given by P (ρ) = ρ γ , γ > 1, κ i , i = 1, 2 are positive constants and ψ(|x − y|) in L[f ] is a communication weight representing a degree of communication between particles located at x and y. For definiteness, we assumed that ψ is uniformly bounded and away from zero, and is sufficiently regular: there exist positive constants ψ m and ψ M such that the communication function ψ satisfies the following properties: Nonnegative constants µ and σ are proportional to the viscosity of the fluid and squared strength of the noise in a random part of the communication weight, respectively. In the sequel, the constants κ 1 , κ 2 and σ will be normalized to the unity for simplicity. There are three mechanisms for flocking particles: the first one is the alignment force which drives particles to flocking state; the second one is from random noise, which is expressed as a degenerate diffusion in PDE level; the last one is the fluid coupling strength i.e., the compressible fluid will interact with the flocking particles through the drag force. We assume some regularity of ψ to guarantee the existence of the coupled system. The positive lower bound of ψ can balance the diffusion and thus the emergence of flocking can be expected. For notational simplicity, we denote the right hand side of (1) 3 by where the shear and bulk viscosity coefficients µ and λ are assumed to satisfy µ = const. > 0, λ(ρ) = ρ β , β > 1, 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 CS-FP type mean-field kinetic equation to describe the motion of flocking particles under random communication [1,20]. On the other hand, the fluid dynamics can be described by various types of hydrodynamic models such as compressible Navier-Stokes (N-S) equation, please refer to [26,27,38] for more details. Accordingly, we will study a coupled system consisting of the compressible N-S equations and the CS-FP equation in the present paper.
Next, we briefly review some earlier results. As mentioned before, coupled systems for the fluid-particles interaction have been used in many contexts, for example, biosprays in medicine [4], sedimentation of solid grain in physics [5,8], fuel-droplets in combustion theory [37] etc. The CS-FP type equation can arise from the kinetic description of the Cucker-Smale flocking ensemble under random communication [1,20], whereas the compressible Navier-Stokes equations have been studied in many references e.g. [23,26,38]. We mention that the global existence of classical solutions to 2-D N-S equations has been established using Caffarelli-Khon-Nirenberg inequality in [27], respectively. There are also extensive mathematical analysis studies on coupled systems. To understand the coupled system between solid particles and incompressible flow, the authors in [19] applied Vlasov-Stokes equations to model the motion of particles with white noise interacting with incompressible viscous stokes flow. In this work, a global-in-time weak solution was constructed in both two and three dimensions. In [24], the authors further considered the nonlinear effect of the fluid and used Navier-Stokes-Vlasov-Fokker-Planck equations to model the particle-fluid coupled system. Under this setting, they showed the asymptotic stability of the equilibrium solutions under small perturbation. Later on, in [10], the global existence of classical solutions to the Euler-Vlasov-Fokker-Planck equations was studied and convergence rate of solutions around the equilibrium state was established as well. Due to loss of diffusion effect in inviscid Euler flow, authors there had to refine the energy estimates and yielded the large time behavior. Moreover, in [36,39], the authors constructed global weak solutions to the Navier-Stokes-Vlasov system without the white noise of the particle. On the other hand, for the coupled system involving compressible fluid in a bounded domain of R 3 , the coupled system of the Vlasov-Fokker-Planck equation with the compressible Navier-Stokes equations was studied in [31,32], in which global weak solutions were constructed and the asymptotic analysis has been done for strong interaction and diffusion effect. Later on, in [16], the authors proved the global existence of classical solutions to the coupled system of Vlasov-Fokker-Planck equation and compressible Euler equations for small initial data. We also refer to related works in [2,3,7].
The main goal of the present paper is to obtain global classical solutions to the coupled system (1) with large initial data. According to the brief review of previous works, for initial data away from equilibrium, the results on global existence of classical solutions are far from being completed. As in [27], in order to extend the classical solutions globally in time, the key point is to obtain the upper bound for fluid density ρ. However, we cannot directly apply the methods employed in [21,27] to derive the upper bound of ρ or high-order energy estimates due to the additional nonlinear coupling terms in (1) 1 and (1) 3 : Due to the complicated and delicate treatment of these coupling terms in the a priori high-order estimates, there are three key ingredients to overcome the difficulties and obtain the upper bound of fluid density: First, we show that the L ∞ (0, T ; L 1 (R 2 ))-norm of the local momentum m k f (x, t) can be controlled by the initial data and the L 1 (0, T ; L k3+2 (R 2 )-norm of fluid velocity u (Lemma 3.2). This is different from the case for the coupled system with the incompressible fluids [20], where the corresponding norm can be controlled by the initial data uniformly. The momentum estimate plays a very important role through the paper since it connects the integrability of the coupling terms with the integrability of fluid velocity u, which is necessary for the energy estimates of the fluid equation. Second, we obtain a modified elementary weighted energy estimate by combining the basic energy estimate, momentum estimate and the Caffarelli-Kohn-Nirenberg inequality, see the estimate of I 22 in Lemma 3.3 for details. This fundamental energy estimate is crucial because it yields the L ∞ (0, T ; L p (R 2 )-integrability of both ρ and m k f (x, t) with p ≥ 1, see Lemma 3.6 and Corollary 1 for details. With these a priori estimates in hand, we can further obtain two different kinds of estimates for ρu L p (p ≥ 2γ), which are essential to the estimate of the commutator ϑ = [u i , R i R j ](ρu j ) and ψ L ∞ in the derivation of upper bound of ρ. Third, with proper observation, we use the uniform boundedness of the particle kinetic energy functional R 4 |v| 2 f dvdx and obtain a modified nonlinear functional Z 2 (t) including the functional R 4 (u − v) 2 f dvdx, which is compatible with the coupled system (1), see Lemma 3.7 for details. The nonlinear functional Z 2 (t) implies that log(1 + ∇ x u L 2 ) can be controlled by ρ 1+βε L ∞ . Finally, combining aforementioned ingredients and estimates, we can apply Brezis-Wainger inequality in Lemma A.6 and the estimates of commutator in Lemma A.7 to obtain the desired upper bound of density ρ (see Lemma 3.9).
The rest of paper is organized as follows. In Section 2, we briefly discuss a framework and present our main results. In Section 3, we provide several lemmas to be used later. In Section 4, we derive a priori estimates in the whole space. In Section 5, we provide a proof of main result. Finally, Section 6 is devoted to a summary of our main results.
Notation. Throughout the paper, C denotes a generic positive constant which may change line by line. The small constants to be chosen are denoted by ε, δ. For function spaces, W k,p (R 2 ) and W k,p (R 4 ) denote the standard Sobolev spaces with standard norm · W k,p , and H k := W k,2 . · p := ( For the special case p = 2, we denote D as D ,2 . 2. Discussion of a framework and main result. In this section, we briefly recall the CS-FP equation, list several elementary inequalities which will be used later, and derive the weighted basic energy estimates of the compressible Navier-Stokes equations and the upper bound of the fluid density ρ. Meanwhile, the estimate of ∇ x u 2 2 will be obtained. Next, we briefly discuss how the kinetic CS-FP equation (1) 1 can be derived from the corresponding stochastic C-S model via the mean-field limit (N → ∞). Let x i and v i be the position and velocity of the i-th particle in R 3 , respectively. Then, the C-S flocking model reads as follows: whereψ(|x j − x i |) denotes the communication weight between the i-th and j-th particles. For a recent survey on the C-S model, we refer to [13]. We now assume that the communication weightψ contains Gaussian white noise in the form of: where η i t is d-dimensional Gaussian white noise. Thus, the C-S model in (3) incorporated with the random communication ansatz in (4) can be rewritten as the stochastic C-S model with a multiplicative noise [1]: where v c t := 1 N N j=1 v j t . In a mean-field limit (N → ∞) [21], the system (5) can be approximated by the following mean-field equation: Throughout the paper, for notational simplicity, we suppress the t-dependence in f : Next, we present our main results whose proof will be given in Section 4.
Theorem 2.1. Suppose that the following conditions hold.

2.
Initial data [f 0 , ρ 0 , u 0 ] satisfy regularities and integrability: Propagation of velocity moments. First, we state elementary energy estimates for the coupled system without proofs.
• Case A.1 (estimate of I 11 ): We use the Hölder inequality and Lemma 3.1 to obtain .
• Case A.2 (estimate of I 12 ): Again we apply the Hölder inequality and the result (i) to obtain • Case A.3 (estimate of I 13 ): We use integration by parts and the estimate |v c | ≤ C(T ) to get In (9), we collect all estimates in Case A.1 -Case A.3 to find d dt Finally, we integrate the above inequality over [0, t] and use the Gronwall lemma to derive the desired estimate.

3.2.
Weighted energy estimates of the fluid variables. In this part, we show the weighted basic energy estimates of the fluid part. This will be achieved by combing the estimates of ρ p (1 ≤ p < +∞). Lemma 3.3. Suppose that initial data set [f 0 , ρ 0 , u 0 ] satisfy the conditions (6), and for a positive constant T ∈ (0, ∞], let [f, ρ, u] be a smooth solution to system (1)- (2) in [0, T ), and the parameters α, γ and m satisfy Then, we have Proof. We multiply (1) 2 by γ ρ γ γ−1 and (1) 3 |x| α u to obtain Next, we provide estimates for I 2i , i = 1, 2.
• Case B.1 (Estimate of I 21 ): As in Lemma 3.2 [27], we have where ε > 0 is a constant which can be arbitrarily small.
844 SEUNG-YEAL HA, BINGKANG HUANG, QINGHUA XIAO AND XIONGTAO ZHANG where we have used the following relation: Note that there exists a positive constant C(α) such that We combine all estimates for I 2i (i = 1, 2) in (11) and choose ε sufficiently small, and then integrate (11) with respect to t over [0, T ] to derive (10).
We apply the operator div to the momentum equation (1) where the effective viscous flux F is defined by On the other hand, consider the following three elliptic problems on the whole space For (13), we can derive the following elliptic estimates in the following lemma.
be solutions to the elliptic problems in (13). Then, we have With Lemma 3.4, we can further derive estimates for u, ψ, η i as follows.
Lemma 3.5. The following estimates hold.
where we have used the relation m = 2 n a and the following interpolation inequality On the other hand, we use (ii) in Lemma 3.1, Lemma 3.2 and (ii) in this lemma to have We combine (14) and (15) to derive a desired estimate (v).
It follows from (13) and (12) 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(x, t) by using (16).
Lemma 3.6. Let β > 1, and assume that the same conditions in Lemma 3.3 hold. Then, we have Proof. We set (h) + to be the positive part of a function h, and we multiply (16) by ρ[(Λ(ρ) − ψ) + ] 2m−1 with m 1 and integrate the resulting relation over R 2 to obtain 1 2m For notational simplicity, we define , and estimate I 3i one by one as follows.
• Case C.1 (estimate of I 31 ) We use (iv) in Lemma 3.5 to have where we have chosen k = β β−1 in the last inequality. • Case C.2 (Estimate of I 32 ): We use (v) in Lemma 3.5 to have 848 SEUNG-YEAL HA, BINGKANG HUANG, QINGHUA XIAO AND XIONGTAO ZHANG and use Lemma 3.5 to have In (18), we collect all estimates to find We set Then, we have On the other hand, by (i) in Lemma 3.5 and Young's inequality, we have Now we choose m sufficiently large to satisfy 1 + 1 2m We use (ii) in Lemma 3.1, (10) and (19) to have We now apply the Gronwall Lemma for (20) using the estimate (ii) in Lemma 3.1 to find (17) 1 . Then (17) 2 follows from (10) and (17) 850 SEUNG-YEAL HA, BINGKANG HUANG, QINGHUA XIAO AND XIONGTAO ZHANG Proof. The estimates in (21) are easy consequences of Lemma 3.2 and Lemma 3.6.
For brevity, we only show the estimate (iii). In fact, we use (i) in Lemma 3.2 for k 1 = 2 and k 2 = 4p − 2 to have Then we further use (ii) in Lemma 3.2 for k = 4p − 2, (3) in Lemma A.4, (ii) in Lemma 3.5 and Lemma 3.6 to have

3.3.
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 ∇ x u 2 2 and ρu p (p ≥ 2γ) to derive the upper bound estimate of the fluid density. We set the material derivative of the fluid velocity byu: and introduce nonlinear functionals: where ε > 0 is a constant which can be arbitrarily small.
Proof. We denote the perpendicular gradient by ∇ ⊥ := (∂ x2 , −∂ x1 ). Then, the momentum equation can be rewritten as follows: We multiply the above identity byu to obtain On the other hand, we use the relations: Now, we estimate the terms I 4i , i = 1, · · · 6, separately.
the elliptic estimates, Sobolev inequality and Corollary 1 to get , Then, the above estimates in (25) yield • Case D.2 (Estimate of I 42 ): By the duality between Hardy H 1 and BMO spaces, we have • Case D.4 (Estimate of I 45 ) From (ii) in Lemma 3.5 and Corollary 1, we have Applying the integration by parts on space variables, we use Corollary 1 to have On the other hand, we apply the integration by parts, and use Lemma 3.1, (3), (21) to have 1 2 This together with (ii) in Lemma 3.1 implies d dt Z 2 (t) + χ 2 (t) ≤C(T )(Z 2 (t) + 1) Then we further use Lemma 3.6 to derive (22).

A GLOBAL SOLVABILITY OF THE 2D COUPLED KINETIC-FLUID MODEL 853
The following ρu p with p ≥ 2γ will play a crucial role in the estimate of the upper bound of the density as in [23].
Proof. Before the proof of (i) in (27), we first derive the estimate of To the end, we multiply the momentum equation by (2 + α )|u| α u to have We integrate the above equation over R 2 , and use integration by parts to obtain Now we estimate I 5i (1 ≤ i ≤ 3) one by one.
• Case E.1 (Estimate of I 51 and I 52 ): Similar to the computation in [29,30], we use (ii) in Lemma 3.5 to have • Case E.2 (Estimate of I 53 ): We use (ii) in Lemma 3.5 and Corollary 1 to have . Therefore, we combine the above three estimates and (28) to have T , we use interpolation inequality to obtain the first result: (ii) By the Hölder inequality, we have We use (ii) in Lemma 3.1 to get Note that Then, we use Lemma A.3 to have On the other hand, we use the Poincaré type inequality in Lemma 3.2 of [18] and get from (ii) and (iv) in Lemma 3.1 that The above two estimates imply Therefore, we substitute the estimates (30) and (31) in (29) to obtain (ii) in (27).
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.6 to have From (ii) in Lemma 3.1 and (i) in Lemma 3.5, we have From the elliptic equation (13) 1 , we use (ii) in Lemma 3.1 and (ii) in (27), Lemma 3.8 to get Collecting the estimates (3.3) and (34), we use (22) to have Now we turn to the estimate of the third term in the RHS of (33). To the end, we first use (i) in Lemma A.5 and (25) 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.7 and (i) in (27) to have

SEUNG-YEAL HA, BINGKANG HUANG, QINGHUA XIAO AND XIONGTAO ZHANG
Then we choose p > 4 sufficiently large and use (35) Finally, we treat the forth term on the RHS of (33). From the elliptic equation (13) 3 , we use (i) in Lemma 3.4 and (v) in Lemma 3.5 for suitably large m to have Finally, we substitute all above estimates into (33) to derive When β > 4 3 , we take positive constant ε sufficiently small to have With Lemma 3.7, 3.9 in hand, we immediately have

Corollary 2. Assume the conditions in Lemma 3.3 hold. Then it holds that
Proof. Note thaṫ Then, we apply the operatoru j [∂ t + div(u·)] to (1) 2,j and use (26) to have • Case F.1 (Estimate of • Case F.2 (Estimate of I 64 and I 65 ): We apply the integration by parts and use (ii) in Lemma 3.5, Corollary 1 and (36) to have , where ε > 0 is a small positive constant. Similarly, we have We collect all estimates of I 6i in (38) to obtain  (24) and (36). We can apply the Gronwall inequality and use (17) to further obtain In order to estimate the first term on the RHS of (39), we apply the operator |x| αu j [∂ t + div(u·)] to (1) 2,j , and use the derivation similar to (38) to obtain For terms I 7i , i = 1, 2, 3, we have Note that we have and Finally, for I 76 , we have Note that for 1 < α < 2 √ 2 − 1, there exists a positive constant C(α) such that . Finally, we collect all estimates of I 7i in (40), combine the resulting inequality and (39), and choose suitably small ε to have Proof. We apply the operator ∇ to the continuity equation (1) 2 , multiply the resulted equation by p|∇ x ρ| p−2 ∇ x ρ, and integrate over R 2 to get • Case G.1 (Estimate of ∇ 2 x u p ): By the interpolation inequality, we use (3) and (37) to have where p, θ and α satisfy the relation: Then we use (3), (17) 1 , Lemma 3.1 and Corollary 2 to have On the other hand, we use Lemma 3.1, (17) 1 and Corollary 2 to have 2 + 1). Then we can further estimate ∇ 2

≤C(T )( ρu
x u p as • Case G.2 (Estimate of ∇ x u ∞ ): By Lemma A.8, we use the estimate of ∇ 2 x u p to have Collecting the estimates of ∇ 2 We apply the Gronwall lemma, and use (17) 2 and Lemma 4.1 to have This inequality together with (44) implies ( Proof. We apply the operator ∂ xi to (1) 3 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 R 4 to give Now we deal with I 8i , i = 1, · · · , 5, as below.
• Case H.1 (Estimate of I 81 ): By direct calculation, we have , where ε > 0 is a small constant. We further estimate I 81 as • Case H.2 (Estimate of I 8i (i = 2, 3, 4): We use integration by parts to obtain We collect all estimates of I 8i in (46) 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 (42) to derive (45).
Proof. First, we rewrite the momentum equation (1) 2 as We multiply the above equation by |x| αu , and integrate the resulting relation with respect to x over R 2 to get Now, we estimate the terms I 9i (1 ≤ i ≤ 4), separately.
• Case I.1 (Estimate of I 9i (1 ≤ i ≤ 3)): As the corresponding estimates in Lemma 4.2 [27], we have • Case I.2 (Estimate of I 94 ): We use (i) in Lemma 3.1 and (i) in Corollary 1 to have

SEUNG-YEAL HA, BINGKANG HUANG, QINGHUA XIAO AND XIONGTAO ZHANG
Note that by (17) Lemma 4.5. Suppose that the conditions in Lemma 3.3 hold. Then, we have where q ≥ 4 α . Proof. By the standard L 2 -estimates for (1) 3 , we use Corollary 2, Lemma 4.1 and Lemma 4.2 to have By the Sobolev inequalities, we use (3) and the above estimate to have Then we can further use Corollary 2, Lemma 4.1, 4.2, 4.4 to obtain and On the other hand, we apply the operator ∇ 2 to the continuity equation (1) 2 and obtain d dt Similarly, we have d dt by Lemma 4.3 for i = 0, 1, 2, 3 and suitably large k.
We set m k ∇ x f := R 2 |v| k |∇ x f |dv. Then, by standard elliptic estimates, we use (32), (50) and Lemma 4.2 to have We combine (48), (49) and (51) to obtain d dt We apply the Gronwall inequality and use Lemma 4.1, 4.2, (63) to have By the continuity equation (1) 2 , we have Then we use (32) and Lemma 4.2 to have Similarly, we apply the operator ∂ t to (52) to obtain Proof. We apply the operator ∂ t to (1) 2 to obtain We multiply the above equation by u tt , and use the integration by parts to obtain As the corresponding estimates in Lemma 3.12 [29], the first term and the second term in the RHS of (53) can be estimated as follows: Now, we turn to treat the third term on the RHS of (53). Rewrite this term as Before the estimation of I 10i , we first note that and µ∆ x u t +∇ x ((µ + λ(ρ))divu t ) (57) ≤C(T )( √ ρu tt 2 + u t 4 + ∇ x u t 2 + ∇ 3 x u 2 + 1), by the estimates in Lemma 4.5.
• Case J.1 (Estimate of I 101 ): By the integration by parts, we use (ii) in Corollary 1, (56) to have 868 SEUNG-YEAL HA, BINGKANG HUANG, QINGHUA XIAO AND XIONGTAO ZHANG • Case J.3 (Estimate of I 103 ): We rewrite I 103 as We apply the integration by parts to use Corollary 1, (56) and (57) to obtain We collect all estimates of I 831 , I 832 in (59) to obtain We combine the estimates of I 8i (1 ≤ i ≤ 3) and (55) to have With the estimates (54) and (60) in hands, we can further estimate (53) as Note that We can further obtain We multiply the above inequality by t, integrate the resulting inequality with respect to t over the interval [τ, t 1 ] with τ, t 1 ∈ [0, T ], and we use the similar arguments as in Lemma 3.12 [29] to obtain On the other hand, we apply operator ∇ 2 to the continuity equation (1) 1 and obtain For P (ρ), we also have To estimate ∇ 2 x u W 1,p on the RHS of (61) and (62), we apply ∂ xi to (1) 3 and obtain µ∆(∂ xi u) + ∇ x ((µ + λ(ρ))div(∂ xi u)) . We combine this estimate and (61), (62) to have d dt Then, by Gronwall's lemma, we have With the bound of ∇ x u L 2 (0,T ;W 2,q ) , q > 2, one can get the second order estimates of f (x, v, t) as follows.
Proof. We apply ∂ α * (|α * | = 2) to the (1) 3 , and multiply the above equation by v kp p|∂ α * f | p−2 ∂ α * f, and integrate the resulted equations with respect to x, v over R 4 to obtain In the sequel, we estimate the terms I 11i separately.
• Case K.1 (Estimate of I 111 ): We rewrite I 111 as follows.
By direct computation, we have Proof. We apply the operator ∂ tt to (1) 2 to obtain ρu ttt + ρu · ∇ x u tt − µ∆ x u tt − ∇ x ((µ + λ(ρ))divu tt ) We mulitply the above equation by t 2 u tt and integrate the resulted equation with respect to x over R 2 to obtain 1 2 In the sequel, we will estimate the terms I 12i , separately.
6. Conclusion. In this paper, we provided a global existence of the coupled system of a kinetic CS-FP equation and compressible N-S equations in the whole space R 2 . Note that the initial data can be arbitrarily large to contain vacuum states. This is motivated by [26,27,29,21]. Compared with the previous results in [27], we need to deal with the drag term in momentum equations (1) 2 additionally. In order to control the drag term, there are three main ingredients in our strategy. First, the global momentum of f can be bounded by t 0 u p dτ with some p > 1 as in Lemma 4.2. Second, by using Caffarelli-Kohn-Nirenberg inequality with best constant, we need to obtain the space weighted estimate of velocity u. Third, the velocity weighted space W 3,p k with some suitably large k is introduced in our proof. Through the weighted energy estimates and elaborate analysis, the global classical solution to the coupled system is established eventually.