GLOBAL WEAK SOLUTION TO THE MAGNETOHYDRODYNAMIC AND VLASOV EQUATIONS

Transcription

1 GLOBAL WEAK SOLUTION TO THE MAGNETOHYDRODYNAMIC AND VLASOV EQUATIONS ROBIN MING CHEN, JILONG HU, AND DEHUA WANG Abstract. An initial-boundary value problem for the fluid-particle system of the inhomogeneous incompressible magnetohydrodynamic equations coupled with the Vlasov equation is studied in a three-dimensional bounded domain. New ideas are introduced to construct the approximate solutions. The existence of global weak solution is established by the energy estimates and the weak convergence method. 1. Introduction Fluid-particle flows arise in a wide range of applications from engineering, medicine, to geophysics and astrophysics ([2, 3, 10, 11, 14, 15, 17, 18, 29, 31, 32]) and have attracted numerous mathematical studies in modeling and analysis. In this paper, we consider a system of partial differential equations describing the motions of uncharged particles in a viscous inhomogeneous incompressible conducting fluid. The system consists of the viscous magnetohydrodynamic (MHD) equations coupled with the Vlasov equation in the following form: ρ t + div(ρu) = 0, (1.1) (ρu) t + div(ρu u) µ u + p = ( H) H m p Ff dv, R 3 (1.2) H t (u H) = (ν H), (1.3) divu = 0, divh = 0, (1.4) f t + v f + div v (Ff) = 0, (1.5) where ρ denotes the density of the fluid, u R 3 the velocity, p the pressure, H R 3 the magnetic field, and f(t, x, v) the density distribution of the solid particles at time t [0, T ] and position x with velocity v R 3 ; µ is the viscosity coefficient, and ν is the magnetic diffusion coefficient; m p is the mass of particles, and F is the drag force. Generally, the drag force depends on the relative velocity u v and the density ρ (e.g. [28]) such as F = F 0 ρ(u v), (1.6) where F 0 is a positive constant. Without loss of generality, we normalize F 0 = m p = 1 throughout the paper Mathematics Subject Classification. 35D30, 35Q35, 76D05, 76W05, 82C40. Key words and phrases. Fluid-particle flows, magnetohydrodynamics, Vlasov equation, weak solutions, weak convergence. 1

2 2 R. M. CHEN, J. HU, AND D. WANG The aim of this paper is to establish the existence of global weak solution to the above system in a bounded domain subject to the following initial conditions: ρ t=0 = ρ 0 (x) 0, x, (1.7) m t=0 = m 0 (x), x, (1.8) H t=0 = H 0 (x), x, (1.9) f t=0 = f 0 (x, v) 0, x, v R 3. (1.10) For the boundary conditions, we consider the non-slip boundary condition for the fluid velocity u, the perfectly conducting wall condition for the magnetic field H, and the purely specular reflection boundary condition for the distribution of particles f, that is, where u = 0, H n = 0, ( H) n = 0; (1.11) f(t, x, v) = f(t, x, v ) for x, v n < 0, (1.12) v = v 2(v n)n, and n is the unit outward normal vector to. We define a linear operator K by Kf(t, x, v) = f(t, x, v ) for x. (1.13) We remark that, in general, the Vlasov equation can be endowed with more general boundary conditions, such as the partially absorbing boundary condition (see [7]). Our main result is also true in those cases. In this paper we only consider the purely specular case to present our ideas avoiding extra tedious details for general boundary conditions. There exists a rich literature on the study of the fluid-particle flows and MHD equations. Concerning the fluid-particle interaction, the existence of the global weak solution to the Stokes-Vlasov equations was first studied by Hamdache [22]. For the Navier-Stokes- Vlasov-Fokker-Planck system (with particle diffusion and fluid compressibility effects), Mellet-Vasseur [27, 28] established the existence of the global weak solution and carried out the asymptotic analysis for the compressible flows in bounded domains. The hydrodynamic limit of the incompressible Navier-Stokes-Vlasov-Fokker-Planck system in both the periodic domain and the whole space settings was studied by Goudon-Jabin-Vasseur [19, 20]. Later, Goudon-He-Moussa-Zhang [18] obtained the existence of classical solutions with small data in incompressible flows. When the particle diffusion is absent, the resulting system becomes the Navier-Stokes-Vlasov system, Boudin-Desvillettes-Grandmont- Moussa [5] obtained global weak solutions for incompressible inhomogeneous flows in a periodic domain. Later the result was extended to general domains by Yu and Wang [33, 34, 35], for both homogeneous and inhomogeneous fluids. The weak solution of a more complicated system namely the Navier-Stokes-Vlasov-Poisson system has been studied by Anoshchenko-Khruslov-Stephan [1]. As for the existence of weak solutions of the MHD system, we refer to Hu-Wang [23], Sermange-Temam [30] and the references therein for the related results. Moreover, concerning the Vlasov equation, many studies have been done in the kinetic theory; see [6, 8, 21, 24, 25] and the references therein. See also [3, 9] and their references for the results on the Vlasov-Fokker-Planck-Euler and Vlasov-Euler systems. The main difficulty of the initial-boundary value problem (1.1)-(1.12) lies in the force term in the Navier-Stokes equation (1.2). The general strategy to construct the global

3 MHD-VLASOV EQUATIONS 3 weak solution to the inhomogeneous Navier-Stokes system, such as in Lions [26], requires the force term to be in L 2 t,x in order to obtain the necessary compactness. However, in our system, for a weak velocity field u, the equation (1.3) of the magnetic field can generate at most L 2 t Hx 1 L t L 2 x regularity for H. Hence we cannot expect to achieve this required regularity for the force term ( H) H. Moreover, the L 2 regularity of the second force term R ρ(u v)f dv cannot be directly obtained from the kinetic 3 theory either. One possible way to get around these issues is to prompt the regularity of the forces by a suitable normalization as in Yu and Wang [33, 34, 35]. But, in our case, we cannot achieve the expected results because of the additional coupling between the fluid and the magnetic filed. In fact, the normalization on the term ( H) H would destroy the energy structure, which is crucially needed for the compactness argument. Another option would be to employ a Galerkin approximation to the whole system, where the coupling terms are regarded as internal forces and thus no exterior force needs to be handled. In this case, as is used in Sermange-Temam [30] for the homogeneous incompressible MHD, the coupling terms do not appear in the energy estimate, which enables one to obtain the necessary compactness. However, the inhomogeneous nature of our system and the coupling with the Vlasov equation share the same difficulty as in the compressible fluid system (see, for instance [16]). More precisely, the Galerkin approximation can only ensure that the solutions are in some Hilbert spaces. But, for ρ and f, we need to show that they have better regularity, for example, ρ C([0, T ]; L p ()) and f L (0, T ; L 1 L ( R 3 )), which are not in a Hilbert space. Our strategy, inspired by the work of Feireisl [16] and Boyer-Fabrie [4], is to employ the Galerkin method only to the fluid and magnetic fields, and to use the Schauder fixed point theorem to couple the density and Vlasov equations with them. In the finite-dimensional spaces for the Galerkin approximation, all norms are equivalent, which is crucial to allow us to circumvent the difficulty caused by the lack of the regularity in the force term in the Navier- Stokes equations. Moreover, the work of DiPerna-Lions [12], Beals-Protopopescu [7] and Hamdache [22] provides the existence, uniqueness, and the estimates on the moments of solutions to the Vlasov equation. Finally an application of the work of DiPerna-Lions- Meyer [13] and Lions [26] gives us the compactness of the approximate solutions. The rest of the paper is organized as follows. In Section 2, we derive some a priori estimates from the equations, introduce the assumptions on the initial data, give the definition of the weak solutions, state our main theorem, and list the preliminary results which will be used in the poof of the main theorem. In Section 3, we set up our approximation scheme and establish the existence of solutions to the approximation problem. Finally in Section 4, we deduce the estimates on the approximate solutions and obtain the compactness results to recover the original system, and hence finish the proof of our main theorem. 2. A Priori Estimates and Main Result Firstly, we consider the energy estimate. In view of (1.1), we multiply (1.2) by u, integrate over and use (1.6) and the boundary condition (1.11) to get 1 d ρ u 2 dx+µ u 2 dx = (( H) H) u dx ρf(u v) u dx. (2.1) 2 dt R 3

4 4 R. M. CHEN, J. HU, AND D. WANG Multiplying (1.3) by H, integrating over, using (1.11) and (1.4), we have 1 d H 2 dx + ν H 2 dx = ( (u H)) H dx. (2.2) 2 dt We remark that, from divh = 0, ( H) = (divh) H = H, thus, we will not distinguish between H 2 dx and H 2 dx in all the energy estimates. For the Vlasov equation, we multiply (1.5) by v 2, integrate over R 3 and integrate by parts to get d f v 2 dvdx (v n) v 2 ρf dvdx dt R 3 R 3 = 2 ρf u v 2 dvdx + 2 ρf(u v)u dvdx. (2.3) R 3 R 3 From the fact v 2 = v 2, and the boundary condition (1.12), we have (v n) v 2 ρf dvdx = (v n) v 2 ρf dvdx + (v n) v 2 ρf dvdx R 3 v n>0 v n<0 = (v n) v 2 ρf dvdx (v n) v 2 ρf dv dx = 0. (2.4) v n>0 v n>0 We can also integrate (1.5) directly to get d f dvdx (v n)ρf dvdx = 0. (2.5) dt R 3 R 3 Similarly as in (2.4), we have (v n)ρf dvdx = (v n)ρf dvdx + (v n)ρf dvdx R 3 v n>0 v n<0 = (v n)ρf dvdx (v n)ρf dv dx = 0, (2.6) v n>0 v n>0 and hence we have the conservation of the mass in the Vlasov equation: d f dvdx = 0. (2.7) dt R 3 Combining (2.1)-(2.7) all together, and using the fact ( (( H) H) u dx = H u H + 1 ) 2 ( H 2 ) u dx (2.8) = ( (u H)) H dx, we have the following energy equality: ( ) d ρ u 2 dx + H 2 dx + f(1 + v 2 )dvdx dt R u 2 dx + 2 H 2 dx + 2 ρf u v 2 dvdx = 0, R 3 (2.9)

5 and its integral form: ρ u 2 dx + + 2µ T 0 T MHD-VLASOV EQUATIONS 5 H 2 dx + f(1 + v 2 )dvdx R 3 T u 2 dxds + 2ν H 2 dxds + 2 ρf u v 2 dvdxds 0 R 3 m 0 2 = dx + H 0 2 dx + f 0 (1 + v 2 )dvdx. ρ 0 R 3 Moreover, since ρ and f satisfy the transport-type equations, we have 0 ρ(t, x) ρ 0 L, 0 f(t, x, v) C(T ) f 0 L, a.e. t, x and v. 0 (2.10) Thus the above estimates motivate us to propose the following assumptions on the initial data: ρ t=0 = ρ 0 L (), ρ 0 0, (2.11) m t=0 = m 0 L 2 (), m 0 = 0 where ρ 0 = 0, m 0 2 ρ 0 L 1 (), (2.12) H t=0 = H 0 L 2 (), divh 0 = 0, (2.13) f t=0 = f 0 L 1 L ( R 3 ), f 0 0, v 3 f 0 L 1 ( R 3 ). (2.14) In order to formulate our problem and the main result, we introduce the following function spaces: V 1 = {u H 1 0 () : divu = 0}, (2.15) H 1 = {u L 2 () : divu = 0 in D }, (2.16) V 2 = {H H 1 () : divh = 0, H n = 0, ( H) n = 0}. (2.17) For the Vlasov part, we also define the boundary sets: and the measure on these sets: γ ± = {(t, x, v) (0, T ) R 3 : ±n v > 0}, dγ = (n v)dvdσdt, where dσ is the standard surface measure of. Define the kth moments of f by m k f(t, x) = f(t, x, v) v k dv, R 3 M k f(t) = m k f dx = f(t, x, v) v k dvdx. R 3 With the above a priori estimates and notations, we define our weak solutions as follows. Definition 2.1. We say (ρ, u, H, f) is a global weak solution to the system (1.1)-(1.5) with the boundary conditions (1.11)-(1.12) and initial conditions (2.11)-(2.14), if, for any

6 6 R. M. CHEN, J. HU, AND D. WANG T > 0, (ρ, u, H, f) satisfies the following properties: ρ 0, ρ L ((0, T ) ), ρ C([0, T ]; L p ()), 1 p <, u L 2 (0, T ; V 1 ()), and ρ u 2 L (0, T ; L 1 ()), H L 2 (0, T ; V 2 ()) L (0, T ; L 2 ()), f 0, f L (0, T ; L ( R 3 ) L 1 ( R 3 )), m 3 f L (0, T ; L 1 ()); (2.18) moreover, for any ζ C 1 ([0, T ] ) with ζ(t, ) = 0, T (ρ t ζ + ρu ζ) dxdt = 0 for any φ C 1 ([0, T ] ) with div x φ = 0 and φ(t, ) = 0, T ( ρu t φ (ρu u) φ + µ u φ) dxdt 0 = T 0 ρ 0 ζ(0, x) dx; (2.19) ( ) (( H) H) ρ f(u v)dv φ dxdt + m 0 φ(0, x) dx; R 3 for any ψ C 1 ([0, T ] ) with ψ(t, ) = 0, T ( H t ψ + (u H) ( ψ) + ν( H) ( ψ)) dxdt 0 = H 0 ψ(0, x) dx; (2.20) (2.21) there exists a trace f + L 1 loc (γ+ ), for any ϕ C 1 ([0, T ] R 3 ) with compact support in v and ϕ(t,, ) = 0, T f( t ϕ + v x ϕ + ρ(u v) v ϕ) dvdxdt 0 R 3 (2.22) = f 0 ϕ(0,, ) dvdx Kf + ϕdγ f + ϕdγ, R 3 γ γ + where K is given in (1.13); and finally, the following energy inequality holds: t ρ u 2 dx + H 2 dx + f(1 + v 2 )dvdx + 2µ u 2 dxds R 3 0 t t + 2ν H 2 dxds + 2 ρf u v 2 dvdxds 0 0 R 3 m 0 2 dx + H 0 2 dx + f 0 (1 + v 2 )dvdx, for a.e. t [0, T ]. ρ 0 R 3 We now state our main theorem. (2.23) Theorem 2.1. Under the assumptions (2.11)-(2.14) on the initial data and boundary conditions (1.11) and (1.12), the initial-boundary value problem to the system (1.1)-(1.5) has a weak solution in the sense of Definition 2.1.

7 MHD-VLASOV EQUATIONS 7 To prove the above Theorem 2.1, we will use some known results which are recalled below. The first lemma is on the Hodge-de Rham type decomposition due to Lions [26]: Lemma 2.1. [26] Let N 2, ρ L (R N ) such that ρ ρ > 0 almost everywhere on R N for some ρ (0, ). Then there exists two bounded operators P δ, Q δ on L 2 (R N ) such that for all m L 2 (R N ), (m p, m q ) = (P ρ m, Q ρ m) is the unique solution in L 2 (R N ) of m = m p + m q, ( ) 1/2 div(ρ 1 m p ) = 0, ( ) 1/2 rot(m q ) = 0. Furthermore, if ρ n L (R N ), ρ ρ n ρ almost everywhere on R N for some 0 < ρ ρ < and ρ n converges almost everywhere to ρ, then (P ρn m n, Q ρn m n ) converges weakly in L 2 (R N ) to (P ρ m, Q ρ m) whenever m n converges weakly to m. Next, we quote the following moment estimates for the Vlasov equation: Lemma 2.2. [22] Let u L p (0, T ; L N+k ()) with 1 p and k 1. Assume that f 0 (L L 1 )( R N ) and v k f 0 L 1 ( R N ). Then, the solution f of (1.5) should have the following estimate: M k f C ( ) N+k (M k f 0 ) 1/(N+k) + ( f 0 L + 1) u L p (0,T ;L N+k ()) for all 0 t T, where the constant C depends only on N and T. We also need the following useful compactness result on the solutions of the Vlasov equation due to DiPerna-Lions-Meyer [13]: Lemma 2.3. [13] Suppose f n t + v xf n = div v (F n f n ) in D ((0, ) R 3 ), where f n is bounded in L (0, ; L 2 ( R 3 )) and f n (1+ v 2 ) is bounded in L (0, ; L 1 ( R 3 F )), n 1+ v is bounded in L ((0, ) R 3 ; L 1 ()). Then R f n η(v) dv is relatively compact in L q (0, T ; L p η 3 ()) for 1 q <, 1 p < 2 and for η such that (1+ v ) L 1 + L, σ σ [0, 2). Finally, we have the following existence and uniqueness of solution to the magnetic equation (1.3) for a given u, similar to [23]: Lemma 2.4. For any given u C(0, T ; C0 2 ()), there exists a unique solution H L 2 (0, T ; V 2 ()) L (0, T ; L 2 ()) to the equation (1.3) satisfying H L (0,T ;H 1 ()) C, H t L 2 ((0,T ) ) C, (2.24) where C only depends on u and the initial data H 0.

8 8 R. M. CHEN, J. HU, AND D. WANG 3. Approximation Scheme In this section, we present the approximation problem, and show the existence of weak solutions to the approximation system. For this purpose, we define a finite dimensional space X k = span{ω i } k i=1, where {ω i } i=1 C 0 () is an orthonormal basis of H 1, given in (2.16), and let Y k = C([0, T ]; X k ). Because we allow the presence of vacuum, we will choose our approximate initial density to be bounded away from zero. We set { ρ 0, in, ρ 0 = 1, in R 3 \, and let (ρ 0 ) ε = ( ρ 0 η ε ), where η is the standard mollifier in R 3. Then the initial density for the approximation system is ρ t=0 = ρ 0,ε = (ρ 0 ) ε + ε. (3.1) Obviously, we have ε ρ 0,ε C, for some universal constant C independent of ε. Moreover we have ρ 0,ε C and ρ 0,ε ρ 0 in L p () for all 1 p <. Next we consider the initial condition for the velocity field u. First, we set and define Obviously, ( ρ 0 ) ε = ( ρ 0 η ε ), m ε 0 = m 0 ρ 1/2 0 ( ρ 0 ) ε. m ε 0 m 0 in L 2 (), and m ε 0(ρ 0,ε ) 1/2 m 0 ρ 1/2 0 in L 2 (). Following Lemma 2.1 we have m ε 0 = ρ 0,ε u 0,ε + q 0,ε, where u 0,ε H 1, q 0,ε L 2 () and q 0,ε = 0 in D. Now we impose the initial condition for the velocity field as u t=0 = u 0,k = P k u 0,ε, (3.2) where P k is the orthogonal projection in H 1 onto X k. Lastly, we impose the initial data for the magnetic field and Vlasov equation as H t=0 = H 0,k = (1 k H 0 ) η k, f t=0 = f 0, (3.3) where 1 k is the characteristic function of k, and k is the sub-domain of away from the boundary by 3/k. With the initial data defined above, our approximation problem can be stated as follows.

9 Definition 3.1. We say (ρ k, u k, H k, f k ), with and MHD-VLASOV EQUATIONS 9 ρ k C([0, T ] ), u k Y k, H k L 2 (0, T ; V 2 ()) L (0, T ; L 2 ()), f k L (0, T ; L 1 L ( R 3 )), is an approximate solution if the following equations are satisfied in the weak sense of Definition 2.1, (ρ k ) t + div(ρ k u k ) = 0, (3.4) (ρ k u k ) t + div(ρ k u k u k ) µ u k + p k = ( H k ) H k ρ k R 3 (u k v)f k dv, (3.5) (H k ) t (u k H k ) = (ν H k ), (3.6) (f k ) t + v x f k + div v ((u k v)ρ k f k ) = 0, (3.7) divu k = 0, divh k = 0, (3.8) with the initial conditions (3.1)-(3.3), boundary conditions (1.11) and (1.12), and the test function space in (2.20) replaced by the restriction on X k. With the above Definition, we have the following existence result. Theorem 3.1. For any T > 0, there exists a global weak solution (ρ k, u k, H k, f k ) to the problem (3.4)-(3.8) with the initial conditions (3.1)-(3.3) and the boundary conditions (1.11) and (1.12). Proof. To solve this approximation problem, we need to apply the Schauder s fixed-point theory, which requires us to construct a map Θ k : Y k Y k. We denote the input of Θ k to be ū k, and the corresponding output Θ k (ū k ) to be u k. Then we construct our map as follows. Step 1. and With the input ū k Y k, we consider the following three linear problems: { (ρk ) t + div((ū k )ρ k ) = 0, (3.9) ρ k t=0 = ρ 0,ε, (H k ) t (ū k H k ) = (ν H k ), divh k = 0, H k (x, 0) = H 0,k, H k n = 0, ( H k ) n = 0, { (fk ) t + v x f k + div v ((ū k v)ρ k f k ) = 0, f k t=0 = f 0, f k (t, x, v) = f k (t, x, v ) on. (3.10) (3.11) For problem (3.9), we employ the classical theory of the transport equation. Since we know div(ū k ) = 0, (3.9) is just a transport equation. We integrate along the characteristic line, i.e. solve the following problem: dx ds = ū k(x, s), X(t ; (x, t)) = x x t [0, T ]. (3.12)

10 10 R. M. CHEN, J. HU, AND D. WANG By the standard theory of transport equation, and the fact that ū k C([0, T ]; C 0 ()), there exists a unique smooth solution X of (3.12) and hence the solution to (3.9) is ρ k (t, x) = ρ 0,ε (X(0 ; (x, t))), for all t [0, T ], x. (3.13) Obviously, we have ε ρ k C. Since ρ 0,ε is smooth, we obtain that ρ k is bounded in C 1 ([0, T ]; C ()). Moreover ρ k is unique. For problem (3.10), from Lemma 2.4 we have the existence and uniqueness of solution H k to (3.10) in L 2 (0, T ; V 2 ()) L (0, T ; L 2 ()) and H k L (0,T ;H 1 ()) C, (3.14) for some constant C depending only on the initial data and the input ū k. More precisely, since X k has finite dimension, all norms in X k are equivalent. This implies that the above C depends on the initial data, T, k, and the L (0, T ; L 2 ()) norm of ū k. For problem (3.11), following the classical theory of Vlasov equation (see [7, 12, 21]) we obtain the unique solution f k in L (0, T ; L 1 L ( R 3 )). We denote its corresponding trace by f + k and we have f + k L (γ + ) C. (3.15) Moreover, from Lemma 2.2, we have the following bound for the third order moment ( 6 M 3 f k C(T ) (M 3 f 0 ) 1/6 + ( f 0 L + 1) ū k L (0,T ;L ())) 6. (3.16) We estimate the lower order moments as the following: m 0 f k = f k dv = f k dv + f k dv R 3 v <r v r C f k L r r 3 v 3 f k dv, (3.17) m 1 f k = v f k dv = R 3 C f k L r r 2 v r v <r v r v f k dv + v r v f k dv v 3 f k dv, (3.18) and m 2 f k = v 2 f k dv = R 3 C f k L r r v <r v r v 2 f k dv + v 2 f k dv v r v 3 f k dv. (3.19) Then, taking r = ( R v 3 f 3 k dv) 1/6 > 0, we have m 0 f k C( f k L + 1)(m 3 f k ) 1/2, (3.20) m 1 f k C( f k L + 1)(m 3 f k ) 2/3, (3.21) m 2 f k C( f k L + 1)(m 3 f k ) 5/6. (3.22)

11 MHD-VLASOV EQUATIONS 11 Since ū k Y k and all norms in X k are equivalent, m 3 f k is bounded in L (0, T ; L 1 ()). This implies m 0 f k L (0,T ;L 2 ()) C, (3.23) m 1 f k L (0,T ;L 2 3 C, (3.24) ()) where C depends on the initial data, T, k and the input ū k. Step 2. We consider the Navier-Stokes equation to solve for our output u k of Θ k. Denote the right hand side in (3.5), with u k replaced by ū k, by G(t, x), G := ( H k ) H k ρ k ū k f k dv + ρ k vf k dv. R 3 R 3 From (3.14), (3.16), (3.20), (3.21) and the fact that ū k Y k is fixed, we know G L (0,T ;L 1 ()) C, (3.25) where C depends on the initial data, T, k and the input ū k. To solve for the output u k, we consider the following weak form of the linearized Navier-Stokes equation: ρ k ((u k ) t + ū k u k ) φ dx + µ u k : φ dx = G φ dx, (3.26) with φ X k. Since X k is a finite dimensional space, we can write k u k = α i (t)ω i. i=1 By using the standard Galerkin method and choosing ω i as the test function φ for each i, we can rewrite (3.26) as M(t)α (t) = A(t)α(t) + B(t), (3.27) where α(t) = (α 1 (t),, α k (t)) R k is the unknown vector, M = (M i,j ) k k and A = (A i,j ) k k are k k matrices, and B = (B 1,, B k ) R k is a vector, defined by the following: M i,j (t) = ρ k ω i ω j dx, A i,j (t) = (ρ k (ū k ω i ) ω j + µ ω i : ω j ) dx, B i (t) = G ω i dx. Since ρ k ε, it follows that M(t) is invertible for any t [0, T ]. From the classical ODE theory, α(t) exists uniquely and is continuous on [0, T ], and the same holds for u k which is defined as Θ k (ū k ), the output of ū k. Step 3. We look for a convex set I, such that Θ k maps I into itself. Consider the following energy inequality, derived by taking φ in (3.26) to be ω i and multiplying the equation by α i, then summing over i from 1 to k and using (3.9) and the fact that all norms in X k being equivalent, ρ(t) u k (t) 2 dx + 1 t t u k 2 dx C(k) G 2 L 2 1 () ds + ρ 0,ε u 0,ε 2 dx, (3.28) 0 0

12 12 R. M. CHEN, J. HU, AND D. WANG where C(k) is a constant which only depends on k. From (3.25), for any constant M > 1 ( ) ρ0,ε u 0,ε ε L 2 () + H 0,k L 2 () + M 0 f 0 + M 2 f 0, we can choose T small enough to make u k (t) L 2 () M for t [0, T ]. With the above M and T, we consider the convex set: { } I := u k C([0, T ]; X k ) : sup u k L 2 () M 0 t T and we see that Θ k maps I to itself. Then we prove the compactness of the map Θ k. We choose φ in (3.26) to be ω i, multiply the equation by α i, and then sum over i from 1 to k to obtain ρ k (u k ) t 2 dx + (ū k u k ) (u k ) t dx + µ u k : ((u k ) t ) dx = G (u k ) t dx. Since all norms in X k are equivalent, and ρ k has a lower bound, we obtain ε (u k ) t 2 dx G (u k ) t dx (ū k u k ) (u k ) t dx µ u k : ((u k ) t ) dx (u k ) t L () G L 1 () + (u k ) t L () ū k L 2 () u k L 2 () + (u k ) t L 2 () u k L 2 () ε 2 (u k) t 2 L 2 () + C(k, ε, µ) ( G 2 L 1 () + ū k 2 L 2 () u k 2 L 2 () + u k 2 L 2 (), ). (3.29) Then, by (3.25), we have the control of the du k dt L (0,T ;L 2 ()). Thus from the Aubin-Lions Lemma, we conclude that Θ k maps I into a compact subset in Y k. What remains to check is the continuity of Θ k. It suffices to prove that the map is sequentially continuous. Suppose that {ū n k } n=0 is a sequence which converges to ū k in Y k strongly. By our definition of Θ k, we denote ρ n k, Hn k and f k n as the corresponding solutions to (3.9), (3.10) and (3.11), and let u n k be Θ(ūn k ). When solving (3.9), we have that {ρn k } is bounded in C 1 ([0, T ]; C()). Thus in light of the Aubin-Lions Lemma, we have that {ρ n k } is pre-compact in C([0, T ] ). For H n k, from Theorem 2.4, we obtain that {Hn k } is bounded in L (0, T ; H 1 ()) and {(H n k ) t} is bounded in L 2 ((0, T ) ). So again the Aubin-Lions Lemma implies that {H n k } is pre-compact in L2 ((0, T ) ). Moreover, {H n k } is also weak* pre-compact in L (0, T ; H 1 ()). As for fk n, from the L bound of fk n, (3.15), (3.23) and (3.24), we have that {fk n} is weak* pre-compact in L ((0, T ) R 3 ), {(fk n)+ } is weak* pre-compact in L (γ + ), {m 0 fk n} is weak* pre-compact in L (0, T ; L 2 ()) and {m 1 fk n} is weak* precompact in L (0, T ; L 3 2 ()). At last for u n k, the control of dun k dt L (0,T ;L 2 ()) can also be obtained. So one more application of the Aubin-Lions Lemma proves that {u n k } is pre-compact in C([0, T ]; L2 ()). Since (3.9), (3.10), (3.11) and (3.26) are all linear problems and the solutions are proved to be unique, we conclude that {u n k } converges to u k = Θ k (ū k ) in Y k, as n. From the Schauder fixed-point theorem, there is a fixed point of Θ k in I, which is a solution to the approximation problem. However, in the above argument, the existence time T may depend on k and ε. So here we need to extend our solution to larger time interval which is independent of k and ε. To do this, consider the energy inequality of the coupled system which has the same form as (2.10). We can show that, for the solution,

13 MHD-VLASOV EQUATIONS 13 u k (t) L 2 () is bounded by 1 ε ( ρ 0,ε u 0,ε L 2 () + H 0,k L 2 () + M 0 f 0 + M 2 f 0 ). Since the norm is bounded, we know that we can extend our solution further in time, and the extension interval does not collapse because the size of the extension interval at each step depends only on the convex set I and the initial norms. Then, after finite times, we can extend the solution to arbitrary large time. So we can safely assume that T does not depend on k and ε. 4. Convergence of the Approximate Problem In this section, we shall finish the proof of our main Theorem 2.1, by passing to the limit of our approximation system as k and ε Pass to the limit as k. Similarly as in the a priori estimates, using a density argument, for any fixed T, we have the following energy inequality: T ρ k u k 2 dx + H k 2 dx + f k (1 + v 2 )dvdx + 2µ u k 2 dxdt R 3 0 T T + 2ν H k 2 dxdt + 2 ρ k f k u k v 2 dvdxdt (4.1) 0 0 R 3 ρ 0,ε u 0,k 2 dx + H 0,k 2 dx + f 0 (1 + v 2 )dvdx. R 3 Here, since we keep ε as a constant, ρ k is still bounded away from zero uniformly. Then we can get the following estimates which are independent of k: The above bounds in turn imply the following convergence: u k L (0,T ;L 2 ()) C, (4.2) H k L (0,T ;L 2 ()) C, (4.3) u k L 2 (0,T ;H0 1 ()) C, (4.4) H k L 2 (0,T ;H 1 ()) C. (4.5) u k u in L 2 (0, T ; H 1 0 ()), H k H in L 2 (0, T ; H 1 ()), ρk u k ρu in L (0, T ; L 2 ()). It turns out that the above convergence is not strong enough for us to pass every term in the system to a limit. We need some extra compactness, which can be achieved as follows. Firstly, we observe that f satisfies a transport-type equation with a reflexive boundary condition, and ρ k has a uniform upper bound in L ((0, T ) ). Using the arguments in [7, 21], we obtain from which we have f k L ((0,T ) R 3 ) C, (4.6) f + k L (γ + ) C, (4.7) f k f in L ((0, T ) R 3 ), (4.8)

14 14 R. M. CHEN, J. HU, AND D. WANG f + k f + in L (γ + ). (4.9) This allows us to pass to the limit for the first two terms and boundary terms in the weak form of the Vlasov equation. Secondly, we deal with the interaction terms between fluid and particles. From (3.16), (3.20), (3.21) and (3.22) and using (4.4), we have where C is independent of k, which implies By (4.6) and an interpolation, one has m 0 f k L (0,T ;L 2 ()) C, (4.10) m 1 f k L (0,T ;L 2 3 C, ()) (4.11) m 2 f k L (0,T ;L 6 C, 5 ()) (4.12) m 0 f k L (0,T ;L 1 ()) C. f k L (0,T ;L 2 ( R 3 )) C. Then, from the above estimates and Lemma 2.3, modulo a subsequence, we have m 0 f k m 0 f in L q (0, T ; L p ()), (4.13) m 1 f k m 1 f in L q (0, T ; L p ()), (4.14) for any 1 q <, 1 p < 2. Next, we apply [26, Theorem 2.4] to obtain some compactness for ρ k and u k. From (4.4), (4.10), (4.11) and the uniform boundness of ρ k, we deduce that ρ k u k m 0 f k and ρ k m 1 f k are bounded in L 2 (0, T ; L 3 2 ()). By (4.3) and (4.4), ( H k ) H k is bounded in L 2 (0, T ; L 1 ()), and hence bounded in L 2 (0, T ; W 1, 3 2 ()). Moreover, since ε ρ k C, by (4.2) and (4.4) we know that ρ k u k u k is bounded in L 2 (0, T ; L 3 2 ()), which implies that div(ρ k u k u k ) is bounded in L 2 (0, T ; W 1, 3 2 ()). Furthermore we know that u k is bounded in L 2 (0, T ; H 1 ). Thus the following estimate on (ρ k u k ) t holds < (ρ k u k ) t, φ > C φ L 2 (0,T ;W 1,3 ()) C φ L 3 (0,T ;W 1,3 ()) (4.15) for any φ D with divφ = 0. This allows us to apply [26, Theorem 2.4] to obtain the following convergence: for any 1 q <, and ρ k ρ in C([0, T ]; L q ()) (4.16) ρk u k ρu in L p (0, T ; L r ()), (4.17) ρ k u k ρu in L p (0, T ; L r ()) (4.18) 6p 3p 4. for 2 < p < and 1 r < Putting together (4.13), (4.14), (4.16) and (4.18), we have the following (u k v)ρ k f k dv R 3 (u v)ρf dv R 3 in L 1 ((0, T ) ), (4.19)

15 MHD-VLASOV EQUATIONS 15 which shows the convergence of the Vlasov force in the Navier-Stokes equations. For the divergence term div v (ρ k (u k v)f k ), using similar argument and Hölder s inequality, we have T T ρ k (u k v)f k v ϕ dvdxdt 0 R 3 0 ρ(u v)f v ϕ dvdxdt R 3 (4.20) for any ϕ D. The above convergence ensures that we can pass to a limit except for the magnetic terms. To deal with these terms, let us write (3.6) as (H k ) t = (u k H k ) (ν H k ). From (4.2)-(4.5), (H k ) t is bounded in L 2 (0, T ; H 1 ()). By the Aubin-Lions Lemma, we have H k H in L 2 ((0, T ) ). (4.21) Obviously, (3.2) implies the strong convergence of u 0,k to u 0,ε in L 2 ((0, T ) ). So the weak form of the limit equations (3.4)-(3.8) holds, as k. Now we consider the limit for the energy inequality. From the convergence obtained above, we can deduce the convergence for all terms except for T f k (1 + v 2 ) dvdx and ρ k f k u k v 2 dvdxdt. R 3 0 R 3 Therefore, it suffices to check the convergence of these two terms. From (4.10) and (4.12), we have, Since is bounded, we have We write T m 0 f k m0 f in L (0, T ; L 2 ()), (4.22) m 2 f k m2 f in L (0, T ; L 6 5 ()). (4.23) f(1 + v 2 ) dvdx lim inf R 3 k R 3 f k (1 + v 2 ) dvdx. (4.24) 0 ρ k f k u k v 2 dvdxdt R 3 T = 0 (ρ k f k u k 2 2ρ k f k u k v + ρ k f k v 2 ) dvdxdt. R 3 (4.25) From (4.13) and (4.17) we have ρ k u k 2 m 0 f k ρ u 2 m 0 f in L 1 ((0, T ) ). (4.26) From (4.14) and (4.18) we have ρ k u k vf k dv ρu vf dv in L R 1 ((0, T ) ). (4.27) 3 R 3 Finally, from (4.23) and (4.16), we have ρ k v R 2 f k dv ρ v 2 f dv in L 1 ((0, T ) ). (4.28) 3 R 3

16 16 R. M. CHEN, J. HU, AND D. WANG Putting all together, we conclude T ρ k f k u k v 2 dvdxdt ρf u v 2 dvdxdt. (4.29) 0 R 3 0 R 3 We note that for the initial value in the energy inequality, by the definition of u 0,k and H 0,k, we have the convergence ρ 0,ε u 0,k 2 dx ρ 0,ε u 0,e 2 dx. (4.30) H 0,k 2 dx H 0 2 dx (4.31) T In summary, we have obtained the following results: Proposition 4.1. For any T > 0, there is a solution (ρ ε, u ε, H ε, f ε ) which satisfies the following system in the weak sense of Definition 2.1: with the initial data ρ t + div(ρu) = 0, (4.32) (ρu) t + div(ρu u) µ u + p = ( H) H ρ (u v)f dv, (4.33) R 3 H t (u H) = (ν H), (4.34) f t + v x f + div v ((u v)ρf) = 0, (4.35) divu = 0, divh = 0, (4.36) ρ t=0 = ρ 0,ε, u t=0 = u 0,ε, H t=0 = H 0, f t=0 = f 0, and the boundary conditions (1.11) and (1.12). In addition, the solution satisfies the following energy inequality: T ρ u 2 dx + H 2 dx + f(1 + v 2 )dvdx + 2µ u 2 dxdt R 3 0 T T + 2ν H 2 dxdt + 2 ρf u v 2 dvdxdt (4.37) 0 0 R 3 ρ 0,ε u 0,ε 2 dx + H 0 2 dx + f 0 (1 + v 2 )dvdx. R Pass to the limit as ε 0. The difficulty here is the degeneracy of the lower bound of ρ, but we can resolve this using [26, Theorem 2.4]. First, consider the initial value of the energy. Since we have q = 0 in D, we have ρ 0,ε u 0,ε 2 1 dx = m ε 0 q 0,ε 2 dx ρ 0,ε ( m ε = q 0,ε 2 2 ) (ρ 0,ε u 0,ε + q 0,ε ) q 0,ε dx ρ 0,ε ρ 0,ε ρ 0,ε ( m ε = 0 2 2u 0,ε q 0,ε q 0,ε 2 ) dx ρ 0,ε ρ 0,ε

17 = MHD-VLASOV EQUATIONS 17 ( m ε 0 2 q 0,ε 2 ) ρ 0,ε ρ 0,ε From (4.37), for each ε, the solution (ρ ε, u ε, H ε, f ε ) also satisfies the following energy inequality: T ρ ε u ε 2 dx + H ε 2 dx + f ε (1 + v 2 ) dvdx + 2µ u ε 2 dxdt R 3 0 T T + 2ν H ε 2 dxdt + 2 ρ ε f ε u ε v 2 dvdxdt (4.38) 0 0 R 3 ρ 0,ε u 0,ε 2 dx + H 0 2 dx + f 0 (1 + v 2 )dvdx R 3 m ε 0 2 dx + H 0 2 dx + f 0 (1 + v 2 )dvdx. ρ 0,ε R 3 Since m ε 0 (ρ 0,ε) 1/2 m 0 ρ 1/2 0 in L 2 (), we have the following bounds: dx. ρ ε u ε L (0,T ;L 2 ()) C, (4.39) H ε L (0,T ;L 2 ()) C, (4.40) u ε L 2 (0,T ;H0 1 ()) C, (4.41) H ε L 2 (0,T ;H 1 ()) C. (4.42) However, u ε L (0,T ;L 2 ()) fails to be bounded now due to the degeneracy of the lower bound of ρ ε, and the L 2 (0, T ; H 1 ()) estimate for (H ε ) t is not available. In this case, from (4.40) and (4.41), we see that u ε H ε is bounded in L 2 (0, T ; L 6 5 ()), which implies that (u ε H ε ) is bounded in L 2 (0, T ; H 2 ()). So we obtain that (H ε ) t is bounded in L 2 (0, T ; H 2 ()). These bounds together with the Aubin-Lions Lemma imply the following convergence: u ε u in L 2 (0, T ; H 1 0 ()), H ε H in L 2 (0, T ; H 1 ()), H ε H in L 2 ((0, T ) ). The above convergence ensures us to pass to the limit as ε 0 for the terms involving H ε, in the distributional sense. To deal with the terms involving f ε, we follow the same track as we did in Subsection 4.1. In particular, the estimates (4.6) and (4.7) also hold, and hence, f ε f in L ((0, T ) R 3 ), (4.43) f + ε f + in L (γ + ). (4.44) Next, since (3.16), (3.20), (3.21) and (3.22) are also true for f ε, from (4.41) we have m 0 f ε L (0,T ;L 2 ()) C, (4.45) m 1 f ε L (0,T ;L 3 C, 2 ()) (4.46) m 2 f ε L (0,T ;L 6 C, 5 ()) (4.47)

18 18 R. M. CHEN, J. HU, AND D. WANG where C is independent of ε. Hence, Lemma 2.2 yields m 0 f ε m 0 f, in L q (0, T ; L p ()), (4.48) m 1 f ε m 1 f, in L q (0, T ; L p ()) (4.49) for any 1 q <, 1 p < 2. Now we estimate (ρ ε u ε ) t. From (4.39), (4.40), (4.41), (4.42), (4.45), and (4.46), we have < (ρ ε u ε ) t, φ > C φ L 2 (0,T ;W 1,3 ()) C φ L 3 (0,T ;W 1,3 ()) (4.50) for any φ D with divφ = 0. Hence, applying [26, Theorem 2.4] we have ρε u ε ρu in L p (0, T ; L r ()), (4.51) for 2 < p < and 1 r < ρ ε u ε ρu in L p (0, T ; L r ()), (4.52) 6p 3p 4, and ρ ε ρ in C([0, T ]; L q ()) (4.53) for any 1 q <. Therefore the following convergence can be obtained and T div(ρ ε u ε u ε ) div(ρu u) in D, (4.54) (u k v)ρ k f k dv (u v)ρf dv in L 1 ((0, T ) ), (4.55) R 3 R 3 ρ k (u k v)f k v ϕ dvdxdt 0 R 3 0 ρ(u v)f v ϕ dvdxdt R 3 (4.56) for any ϕ D. For the initial values in the Naiver-Stokes equations, we observe that, by the decomposition and the convergence of m ε 0, ρ 0,ε u 0,ε φ dx = m ε 0 φ dx m 0 φ dx (4.57) for any φ D such that divφ = 0. Therefore, we have obtained the convergence of all terms in the weak form of (4.32)-(4.36). Finally we consider the limit of the energy inequality. From (4.45) and (4.47), we have, T m 0 f ε m0 f in L (0, T ; L 2 ()), (4.58) m 2 f ε m2 f in L (0, T ; L 6 5 ()). (4.59) By (4.49), (4.51), (4.52) and (4.53), we obtain f(1 + v 2 ) dvdx lim inf f ε (1 + v 2 ) dvdx, R 3 ε R 3 ρ ε u ε 2 m 0 f ε ρ u 2 m 0 f in L 1 ((0, T ) ), ρ ε u ε vf ε dv ρu vf dv in L R 1 ((0, T ) ), 3 R 3 ρ ε v R 2 f ε dv ρ v 2 f dv in L 1 ((0, T ) ), 3 R 3

19 MHD-VLASOV EQUATIONS 19 which, together with the convergence of initial data, yields the limiting energy inequality. The proof of Theorem 2.1 is complete. Acknowledgments R.M. Chen s research was supported in part by the National Science Foundation under Grant DMS and by the University of Pittsburgh CRDF grant. J. Hu s research was supported in part by the National Science Foundation under Grants DMS and DMS D. Wang s research was supported in part by the National Science Foundation under Grant DMS References [1] O. Anoshchenko, E. Khruslov, H. Stephan, Global Weak Solutions to the Navier-Stokes-Vlasov-Poisson System. Zh. Mat. Fiz. Anal. Geom. 6 (2010), no. 2, , 247, 249. [2] C. Baranger, L. Boudin, P.-E. Jabin, and S. Mancini. A modeling of biospray for the upper airways. ESAIM: Proc. 14 (2005) [3] C. Baranger, L. Desvillettes, Coupling Euler and Vlasov equations in the context of sprays: the localin-time, classical solutions. J. Hyperbolic Differ. Equ. 3 (2006), no. 1, [4] F. Boyer, P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models. Appl. Math. Sci Springer, New York, [5] L. Boudin, L. Desvillettes, C. Grandmont, A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations. Differential Integral Equations 22 (2009), no , [6] M. Bostan, Weak Solutions for the Vlasov-Poisson Initial-Boundary Value Problem with Bounded Electric Field Chin. Ann. Math. Ser. B 28 (2007), no. 4, [7] R. Beals, V. Protopopescu, An abstract time dependant transport equations. J. Math. Anal. Appl., 121 (1987), [8] J. A. Carrillo, Global Weak solutions for the Initial-Boundary-Value Problems to the Vlasov-Poisson- Fokker-Planck System. Math. Methods Appl. Sci. 21 (1998), no. 10, [9] J. A. Carrillo, R. Duan, A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Fokker- Planck-Euler system. Kinetic and Related Models 4 (2011), no. 1, [10] R. Caflisch, G. C. Papanicolaou, Dynamic theory of suspensions with brownian effects. SIAM J. Appl. Math. 43 (1983), no.4, [11] K. Domelevo, Long time behavior for a kinetic modeling of two phase flows with thin sprays and point particles. Preprint de TMR-project Asymptotics Methods in Kinetic Theory (2001). [12] R. J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, [13] R. J. DiPerna, P.-L. Lions, Y. Meyer, L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 3 4, [14] K. Domelevo, J.-M. Roquejoffre, Existence and stability of travelling waves solutions in a kinetic model of two phase flows. Comm. Part. Diff. Eq. 24 (1999), no. 1 2, [15] K. Domelevo, M.-H. Vignal, Limits visqueuses pour des systèmes de type Fokker-Planck-Burgers unidimensionnels. C. R. Acad. Sci. Paris. Sér. I Math. 332 (2001), no. 9, [16] E. Feireisl, Dynamics of viscous compressible fluids. Oxford Lecture Ser. Math. Appl. Oxford University Press, Oxford, [17] T. Gemci, T. E. Corcoran, and N. Chigier. A numerical and experimental study of spray dynamics in a simple throat model. Aerosol Sci. Technol, 36 (2002), [18] T. Goudon, L. He, A. Moussa, and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equillibrium. SIAM J. Math. Anal. 42 (2010), no. 5, [19] T. Goudon, P.-E. Jabin, A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime. Indiana Univ. Math. J. 53 (2004), no. 6,

20 20 R. M. CHEN, J. HU, AND D. WANG [20] T. Goudon, P.-E. Jabin, A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime. Indiana Univ. Math. J. 53 (2004), no. 6, [21] Y. Guo, Global Weak Solutions of the Vlasov-Maxwell System with Boundary Conditions Comm. Math. Phys. 154 (1993), no. 2, [22] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations. Japan J. Indust. Appl. Math., 15 (1998), no. 1, [23] X. Hu, D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Rational Mech. Anal. 197 (2010), no. 1, [24] H. J. Hwang, Regularity for the Vlasov-Poisson System in a Convex Domain SIAM J. Math. Anal. 36 (2004), no. 1, [25] S. Mischler, On the Initial Boundary Value Problem for the Vlasov-Poisson-Boltzmann System Comm. Math. Phys. 210 (2000), no. 2, [26] P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Ser. Math. Appl. 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, [27] A. Mellet, A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math. Models Methods Appl. Sci. 17 (2007), no. 7, [28] A. Mellet, A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Comm. Math. Phys. 281 (2008), no. 3, [29] P. J. O Rourke, Collective drop effects on vaporizing liquid sprays. PhD thesis, Los Alamos National Laboratory, [30] M. Sermange, R. Temam, Some Mathematical Questions Related to the MHD Equations. Comm. Pure Appl. Math. (1983), Vol. XXXVI, , [31] W. E. Ranz, W. R. Marshall, Evaporization from drops, part I-II. Chem. Eng. Prog. 48 (1952), no. 3, [32] F. A. Williams, Combustion theory. Benjamin Cummings, [33] C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl. (9) 100 (2013), no. 2, [34] C. Yu, Global well-posedness for the two dimensional Navier-Stokes-Vlasov Equations, preprint, 2014, arxiv: [35] D. Wang, C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov Equations. J. Differential Equation 259 (2015), no. 8, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA address: mingchen@pitt.edu Department of Mathematics, University of Pittsburgh, Pittsburgh, PA address: jih62@pitt.edu Department of Mathematics, University of Pittsburgh, Pittsburgh, PA address: dwang@math.pitt.edu