GLOBAL EXISTENCE AND LARGE-TIME BEHAVIOR OF SOLUTIONS TO THE THREE-DIMENSIONAL EQUATIONS OF COMPRESSIBLE MAGNETOHYDRODYNAMIC FLOWS

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1 GLOBAL EXISTENCE AND LARGE-TIME BEHAVIOR OF SOLUTIONS TO THE THREE-DIMENSIONAL EQUATIONS OF COMPRESSIBLE MAGNETOHYDRODYNAMIC FLOWS XIANPENG HU AND DEHUA WANG Abstract. The three-dimensional equations of compressible magnetohydrodynamic isentropic flows are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of global weak solutions are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent γ > 3 and constant viscosity coefficients Introduction Magnetohydrodynamics MHD concerns the motion of conducting fluids in an electromagnetic field with a very broad range of applications. The dynamic motion of the fluid and the magnetic field interact strongly on each other. The hydrodynamic and electrodynamic effects are coupled. The equations of three-dimensional compressible magnetohydrodynamic flows in the isentropic case have the following form [2, 18, 19]: ρ t + divρu =, ρu t + div ρu u + p = H H + µ u + λ + µ divu, 1.1 H t u H = ν H, divh =, where ρ denotes the density, u R 3 the velocity, H R 3 the magnetic field, pρ = aρ γ the pressure with constant a > and the adiabatic exponent γ > 1; the viscosity coefficients of the flow satisfy 2µ + 3λ > and µ > ; ν > is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, and all these kinetic coefficients and the magnetic diffusivity are independent of the magnitude and direction of the magnetic field. The symbol denotes the Kronecker tensor product. Usually, we refer to the first equation in 1.1 as the continuity equation, and the second equation as the momentum balance equation. It is well-known that the electromagnetic fields are governed by the Maxwell s equations. In magnetohydrodynamics, the displacement current can be neglected [18, 19]. As a consequence, the last equation in 1.1 is called the induction equation, and the electric field can be written in terms of the magnetic field H and the velocity u, E = ν H u H. Although the electric field E does not appear in 1.1, it is indeed induced according to the above relation by the moving conductive flow in the magnetic field. In this paper, we are interested in the global existence and large-time behavior of solutions to the three-dimensional MHD equations 1.1 in a bounded domain R 3 with 1991 Mathematics Subject Classification. 35Q36, 35D5, 76W5. Key words and phrases. Three-dimensional magnetohydrodynamics MHD equations, global solutions, large-time behavior, weak convergence, renormalized solutions. 1

2 2 XIANPENG HU AND DEHUA WANG the following initial-boundary conditions: ρx, = ρ x L γ, ρ x, ρx, ux, = m x L 1, m = if ρ =, Hx, = H x L 2, divh = in D, u =, H =. m 2 ρ L 1, 1.2 There have been a lot of studies on MHD by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges; see [3, 4, 5, 7, 12, 15, 19, 25] and the references cited therein. In particular, the one-dimensional problem has been studied in many papers, for examples, [3, 4, 7, 15, 17, 22, 25] and so on. However, many fundamental problems for MHD are still open. For example, even for the one-dimensional case, the global existence of classical solution to the full perfect MHD equations with large data remains unsolved when all the viscosity, heat conductivity, and diffusivity coefficients are constant, although the corresponding problem for the Navier-Stokes equations was solved in [16] long time ago. The reason is that the presence of the magnetic field and its interaction with the hydrodynamic motion in the MHD flow of large oscillation cause serious difficulties. In this paper we consider the global weak solution to the three-dimensional MHD problem with large data, and investigate the fundamental problems such as global existence and large-time behavior. A multi-dimensional nonisentropic MHD system for gaseous stars coupled with the Poisson equation is studied in [5], where all the viscosity coefficients depend on temperature, and the pressure depends on density asymptotically like the isentropic case pρ = aρ 5 3. In this paper, we study the multi-dimensional isentropic problem with γ > 3 2, where all the viscosity coefficients µ, λ, ν are constant. We remark that γ = 5 3 for the monoatomic gases. When there is no electromagnetic field, system 1.1 reduces to the compressible Navier- Stokes equations. See [9, 14, 21] and their references for the studies on the multi-dimensional Navier-Stokes equations. In particular, to overcome the difficulties of large oscillations of solutions, especially of density, the concept of a renormalized solutions is used in [21, 9]. Based on this idea, we study the initial-boundary value problem for the MHD system in a bounded three-dimensional domain. The goal of this paper is to establish the existence of global weak solutions for large initial data in certain functional spaces for γ > 3 2 and to study the large-time behavior of global weak solutions when the magnetic field and interaction present. The existence of global weak solutions is proved by using the Faedo-Galerkin method and the vanishing viscosity method. We first obtain a priori estimates directly from 1.1, which is the backbone of our result. In the proof of the existence, we use the similar approximation scheme to that in [1] which consists of Faedo-Galerkin approximation, artificial viscosity, and artificial pressure. Then, motivated by the work in [6], we show that an improvement on the integrability of density can ensure the effectiveness and convergence of our approximation scheme. More specifically, we show that the uniform bound of ρ γ ln1 + ρ in L 1, rather than the uniform bound of ρ γ+θ in L 1 for some θ > as used in [1, 9, 21], ensures the vanishing of artificial pressure and the strong convergence of the density. To overcome the difficulty arising from the possible large oscillations of the density ρ, we adopt the method in Lions [21] and Feireisl [9] which is based on the celebrated weak continuity of the effective viscous flux p λ+2µdivu see also Hoff [13]. The estimates obtained by our approach produce further the large-time behavior of the global weak solutions to the initial-boundary value problem To achieve our goal for the MHD problem, we also need to develop estimates to deal with the magnetic field and its coupling and interaction with the fluid variables. The nonlinear term H H will be dealt with by the idea arising in incompressible Navier-Stokes equations.

3 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 3 We organize the rest of the paper as follows. In Section 2, we derive a priori estimates from 1.1, give the definition of the weak solutions, and also state our main results and give our approximation scheme. In Section 3, we will show the unique solvability of the magnetic field in terms of the velocity field. In Section 4, following the method in [9] and the result obtained in Section 3, we show the existence of solutions to the approximation system. In Section 5, we follow the technique in [1] with some modifications to get the strong convergence of ρ in L 1, T. In Section 6, motivated by [11] we study the large-time behavior of global weak solutions to Main Results In this section, we reformulate the initial-boundary value problem and state the main results. We first formally derive the energy equation and some a priori estimates. Multiplying the second equation in 1.1 by u, integrating over, and using the boundary condition in 1.2, we obtain d dt = 1 2 ρu2 + a γ 1 ργ dx + µ Du 2 + λ + µdivu 2 dx H H u dx. The term on the right hand side of 2.1 can be rewritten as H H u dx = H u H H 2 u dx. Hence, 2.1 becomes d 1 dt 2 ρu2 + a γ 1 ργ = H u H H 2 u dx + µ Du 2 + λ + µdivu 2 dx dx Multiplying the third equation in 1.1 by H, integrating over, and using the boundary condition in 1.2 and the condition divh =, one has 1 d H 2 dx + ν H H dx = u H H dx dt Direct calculations show that ν H H dx = ν Thus 2.3 yields 1 d H 2 dx + ν 2 dt u H H dx = H 2 dx = H 2 dx, H u H H 2 u dx. H u H H 2 u dx. 2.4 Adding 2.2 and 2.4 gives d 1 dt 2 ρu2 + a γ 1 ργ H 2 dx + µ Du 2 + λ + µdivu 2 + ν H 2 dx =. 2.5

4 4 XIANPENG HU AND DEHUA WANG From Lemma 3.3 in [23], our assumptions on initial data, and 2.5, we have the following a priori estimates: ρ u 2 L [, T ]; L 1 ; u L 2 [, T ]; H 1 ; H L 2 [, T ]; H 1 L [, T ]; L 2 ; ρ L [, T ]; L γ. Multiplying the continuity equation i.e., the first equation in 1.1 by b ρ, we obtain the renormalized continuity equation: bρ t + divbρu + b ρρ bρdivu =, 2.6 for some suitable function b C 1 R +. Following the strategy in [21, 9], we introduce the concept of finite energy weak solution ρ, u, H to the initial-boundary value problem in the following sense: The density ρ is a non-negative function, ρ C[, T ]; L 1 L [, T ]; L γ, ρx, = ρ, and the momentum ρu satisfies 2γ γ+1 ρu C[, T ]; Lweak ; The velocity u and the magnetic field H satisfy the following: u L 2 [, T ]; H 1, H L 2 [, T ]; H 1 C[, T ]; L 2 weak, ρu u, u H, and H H are integrable on, T, and ρux, = m, Hx, = H, divh = in D ; The system 1.1 is satisfied in D R 3, T provided that ρ, u, and H are prolonged to be zero outside ; The continuity equation in 1.1 is satisfied in the sense of renormalized solutions, that is, 2.6 holds in D, T for any b C 1 R + satisfying b z = for all z R + large enough, say, z z, 2.7 where the constant z depends on the choice of function b; The energy inequality t Et + µ Du 2 + λ + µdivu 2 + ν H 2 dxds E, holds for a.e t [, T ], where 1 Et = 2 ρu2 + a γ 1 ργ H 2 dx, and E = 1 m 2 + a 2 ρ γ 1 ργ H 2 dx. Remark 2.1. As a matter of fact, the function b does not need to be bounded. By Lebesgue Dominated convergence theorem, we can show that if ρ, u is a pair of finite energy weak solutions in the renormalized sense, they also satisfy 2.6 for any b C 1, C[, satisfying b zz cz γ 2 for z larger than some positive constant z. 2.8 Now our main result on the existence of finite energy weak solutions reads as follows.

5 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 5 Theorem 2.1. Assume that R 3 be a bounded domain with a boundary of class C 2+κ, κ >, and γ > 3 2. Then for any given T >, the initial-boundary value problem has a finite energy weak solution ρ, u, H on, T. Remark 2.2. The fluid density ρ as well as the momentum ρu should be recognized in the sense of instantaneous values cf. Definition 2.1 in [9] for any time t [, T ]. As a direct application of Theorem 2.1, we have the following result on the large-time behavior of solutions to the problem : Theorem 2.2. Assume that ρ, u, H is the finite energy weak solution to obtained in Theorem 2.1, then there exist a stationary state of density ρ s which is a positive constant, a stationary state of velocity u s =, and a stationary state of magnetic field H s = such that, as t, ρx, t ρ s strongly in L γ ; ux, t u s = strongly in L 2 ; 2.9 Hx, t H s = strongly in L 2. The proof of Theorem 2.1 is based on the following approximation problem: ρ t + divρu = ε ρ, ρu t + div ρu u + a ρ γ + δ ρ β + ε u ρ = H H + µ u + λ + µ divu, H t u H = ν H, divh =, 2.1 with the initial-boundary conditions which will be specified in Section 4, where β > is a constant to be determined later, and ε >, δ >. Taking ε and δ in 2.1 will give the solution of 1.1 in Theorem 2.1. We remark that the nonlinear term H H can be dealt with by the idea arising in incompressible Navier-Stokes equations, but there are no estimates good enough to control possible oscillations of the density ρ. In order to overcome this difficulty, we adopt the method in Lions [21] and Feireisl [9] which is based on the celebrated weak continuity of the effective viscous flux p λ + 2µdivu. More specifically, it can be shown that aρ γ n λ + 2µdivu n bρ n aρ γ λ + 2µdivubρ weakly in L 1, T, where ρ n and u n are a suitable sequence of approximate solutions, and the symbol F v stands for a weak limit of {F v n } n=1. Remark 2.3. Similarly to [8], our approach also works for the general barotropic flow: pρ = aρ γ + zρ, zρ lim ρ ρ γ [,, where a >, γ > 3 2. Moreover, we also can extend our results to the initial-boundary value problem for 1.1 in an exterior domain by using the method of invading domain cf. Section 7.11 in [24] and the following special type of Orlicz spaces L p q see Appendix A in [21]: L p q = { f L 1 loc : f { f <η} L q, and f { f η} L p, for some η > }.

6 6 XIANPENG HU AND DEHUA WANG 3. The Solvability of The Magnetic Field In order to prove the existence of solutions to 2.1 by Faedo-Galerkin method, we need to show that the following system can be uniquely solved in terms of u: H t u H = ν H, divh =, 3.1 Hx, = H, H =. In fact, we have the following properties: Lemma 3.1. Let u C[, T ]; C 2 ; R 3. Then there exists at most one function H L 2 [, T ]; H 1 L [, T ]; L 2 which solves 3.1 in the weak sense on, T, and satisfies boundary and initial conditions in the sense of traces. Proof. Let H 1, H 2 be two solutions of 3.1 with the same data. Then we have H 1 H 2 t u H 1 H 2 = ν H 1 H Multiplying 3.2 by H 1 H 2, integrating over, and using the Cauchy-Schwarz inequality, we obtain 1 d H 1 H 2 2 dx + ν H 1 H 2 2 dx 2 dt = u H 1 H 2 H 1 H 2 dx ν H 1 H 2 2 dx + 1 u H 1 H 2 2 dx 2 2ν ν H 1 H 2 2 dx + Cν, u H 1 H 2 2 dx. 2 This implies 1 d H 1 H 2 2 dx + ν H 1 H 2 2 dx 2 dt 2 Cν, u H 1 H 2 2 dx. Then, Lemma 3.1 follows directly from Gronwall s inequality and Lemma 3.2. Let R 3 be a bounded domain of class C 2+κ, κ >. Assume that u C[, T ]; C 2 ; R 3 is a given velocity field. Then the solution operator u H[u] assigns to u C[, T ]; C 2 ; R 3 the unique solution H of 3.1. Moreover, the solution operator u H[u] maps bounded sets in C[, T ]; C 2 ; R 3 into bounded subsets of Y := L 2 [, T ]; H 1 L [, T ]; L 2, and the mapping u C[, T ]; C 2 ; R 3 H Y is continuous on any bounded subsets of C[, T ]; C 2 ; R 3. Proof. The uniqueness of the solution to 3.1 is a consequence of Lemma 3.1. Noticing that H = divh H,

7 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 7 then 3.1 becomes H t u H = ν H, divh =, Hx, = H, H =, 3.4 which is a linear parabolic-type problem in H, so the existence of solution can be obtained by the standard Faedo-Galerkin methods. And from 3.3, we can conclude that the solution operator u H[u] maps bounded sets in C[, T ]; C 2 ; R 3 into bounded subsets of the set Y = L 2 [, T ]; H 1 L [, T ]; L 2. Our next step is to show the solution operator is continuous from any bounded subset of C[, T ]; C 2 ; R 3 to L 2 [, T ]; H 1 L [, T ]; L 2. To this end, let {u n } n=1 be a bounded sequence in C[, T ]; C 2 ; R 3, i.e., {u n } n=1 B, K C[, T ]; C 2 ; R 3 for some K >, and u n u in C[, T ]; C 2 ; R 3, as n. Then, we have, denoting H[u] by H u, and H[u n ] by H n, 1 d H n H u 2 dx + ν H n H u 2 dx 2 dt = u n H n u H u H n H u dx u n u H n + u H n H u H n H u dx u n u H n 2 Y + K H n H u L 2 H n H u H 1 C 2 u n u + c H n H u 2 L 2 + ν 2 H n H u 2 H 1, where, we used the fact that H n [u n ] is bounded in Y, says, by C = CK, ν. This implies that 1 d H n H u 2 dx + ν H n H u 2 dx 2 dt C 2 u n u + c H n H u 2 L 2 Integrating 3.5 over t, T, and then taking the upper limit over n on the both sides, we get, noting that u n u in C[, T ]; C; 2 R 3, 1 lim sup H n H u 2 dx + ν t lim sup H n H u 2 dxds 2 n 2 n t t 3.6 c lim sup H n H u 2 L 2 ds c lim sup H n H u 2 L 2 ds, n n thus, from 3.6, using Gronwall s inequality and the same initial value for H n and H u, we get lim sup H n H u 2 tdx =. n This yields, from 3.6 again, t lim sup H n H u 2 dxds =. n Therefore, we obtain H n H u in Y. This completes the proof of the continuity of the solution operator.

8 8 XIANPENG HU AND DEHUA WANG 4. The Faedo-Galerkin Approximation Scheme In this section, we establish the existence of solutions to 2.1 following the approach in [1] with the extra efforts to overcome the difficulty arising from the magnetic field. Let X n = span{η j } n j=1 be the finite-dimensional space endowed with the L 2 Hilbert space structure, where the functions η j D; R 3, j = 1, 2,..., form a dense subset in, says, C; 2 R 3. Through this paper, we use D to denote C, and D for the sense of distributions. The approximate velocity field u n C[, T ]; X n satisfies the system of integral equations: ρu n x, t η dx m,δ η dx t = µ u n divρu n u n + λ + µdivu n aρ γ δρ β ε ρ u n + H H η dxdτ, 4.1 for any t [, T ] and any η X n, where ε, δ, and β are fixed positive parameters. The density ρ n = ρ[u n ] is determined uniquely as the solution of the Neumann initial-boundary value problem cf. Lemma 2.2 in [1]: ρ t + divρu n = ε ρ, ρ n =, 4.2 ρx, = ρ,δ x; and the magnetic field H n = H[u n ] as a solution to the system 3.1. The initial data ρ,δ is a smooth function in C 3 satisfying the homogeneous Neumann boundary conditions ρ,δ n =, and moreover, we set < δ ρ,δ δ 1 2β, 4.3 ρ,δ ρ in L γ, {ρ,δ < ρ } for δ ; 4.4 m,δ x = { m x, if ρ,δ ρ x,, if ρ,δ < ρ x. 4.5 Due to Lemma 3.1 and Lemma 3.2, the problem 4.1, 4.2, and 3.1 can be solved locally in time by means of the Schauder fixed-point technique; see for example Section 7.2 of Feireisl [9]. As in [1, 9] the role of the artificial pressure term δρ β in 4.1 is to provide additional estimates on the approximate densities in order to facilitate the limit passage ε cf. Chapter 7 in [9]. To this end, one has to take β large enough, says, β > 8, and to re-parametrize the initial distribution of the approximate densities so that δ ρ β,δ dx as δ. 4.6 To obtain uniform bounds on u n, we derive an energy equality similar to 2.5 as follows. Taking ηx = u n x, t with fixed t in 4.1 and repeating the procedure for a priori estimates in Section 2, we deduce a kinetic energy equality : d 1 dt 2 ρ nu 2 n + a γ 1 ργ n + δ β 1 ρβ n H n 2 dx + µ Dun 2 + λ + µdivu n 2 + ν H n H n dx ε aγρ γ 2 n + δβρ β 2 n ρn 2 dx =.

9 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 9 The uniform estimates obtained from 4.7 furnish the possibility to repeat the above fixed point argument to extend the local solution u n to the whole time interval [, T ]. Then, by the solvability of equations 4.2 and 3.1, we obtain the functions {ρ n, H n } on the whole time interval [, T ]. The next step in the proof of Theorem 2.1 consists of passing to the limit as n in the sequence of approximate solutions {ρ n, u n, H n } obtained above. We first observe that the terms related to u n and ρ n can be treated similarly to Section in [9], due to the energy equality 4.7. It remains to show the convergence of the sequence of solutions {H n } n=1. From 4.7, we conclude H n is bounded in L [, T ]; L 2 L 2 [, T ]; H 1. This implies that, by the compactness of H 1 L 2 and selecting a subsequence if necessary, there exists a function H L [, T ]; L 2 L 2 [, T ]; H 1 with divh = such that H n, t H, t in L 2 for a.e. t [, T ]. Thus, Similarly, we have H n H n H H in D, T. u n H n u H in D, T, where u n u weakly in L 2 [, T ]; H 1. Therefore, 4.1 and 3.1 holds at least in the sense of distribution. Moreover, by the uniform estimates on u, H and the third equation in 1.1, we know that the map t H n x, tϕx dx for any ϕ D, is equi-continuous on [,T]. By the Ascoli-Arzela Theorem, we know that t Hx, tϕx dx is continuous for any ϕ D. Thus, H satisfy the initial condition in 3.1 in the sense of distribution. Now, we are ready to summarize an existence result for problem 4.1, 4.2, and 3.1 cf. Proposition 7.5 in [9]. Proposition 4.1. Assume that R 3 is a bounded domain of the class C 2+κ, κ >. Let ε >, δ >, and β > max{4, γ} be fixed. Then for any given T >, problem 4.1, 4.2, and 3.1 admits at least one solution ρ, u, H in the following sense: 1 The density ρ is a non-negative function such that ρ L r [, T ]; W 2,r, t ρ L r, T, for some r > 1, the velocity u belongs to the class L 2 [, T ]; H 1, equation 4.2 holds a.e on, T, and the boundary condition as well as the initial data condition on ρ are satisfied in the sense of traces. Moreover, the total mass is conserved, especially, ρx, t dx = ρ,δ x dx, for all t [, T ]; and the following estimates hold: T δ ρ β+1 dxdt Cε, ε T ρ 2 dxdt C with C independent of ε.

10 1 XIANPENG HU AND DEHUA WANG 2 All quantities appearing in equation 4.1 are locally integrable, and the equation is satisfied in D, T. Moreover, we have γ+1 ρu C[, T ]; Lweak, and ρu satisfies the initial condition. 3 All terms in 3.1 are locally integrable on, T. The magnetic field H satisfies equation 3.1 and the initial data in the sense of distribution. divh = also holds in the sense of distribution. 4 The energy inequality 1 2 ρu2 + a γ 1 ργ + δ β 1 ρβ H 2 x, t dx + µ Du 2 + λdivu 2 + ν H 2 x, t dx + ε aγρ γ 2 + δβρ β ρ 2 x, t dx 1 m 2 + a 2 ρ γ 1 ργ + δ β 1 ρβ H 2 dx, holds a.e t [, T ]. In the next section, we will complete the proof of Theorem 2.1 by taking vanishing artificial viscosity and vanishing artificial pressure. 2γ 5. The Convergence of the Approximate Solution Sequence Now we have the approximate solutions {ρ ε,δ, u ε,δ, H ε,δ } obtained in Section 4. To prove Theorem 2.1 we need to take the limits as the artificial viscosity coefficient ε and as the artificial pressure coefficient δ. First, following Chapter 7 in [9] see also [1], we can pass to the limit as ε to obtain the following result: Proposition 5.1. Assume R 3 is a bounded domain of class C 2+κ, κ >. Let δ >, and 6γ β > max{4, 2γ 3 } be fixed. Then, for given initial data ρ, m as in and H as in 1.2, there exists a finite energy weak solution ρ, u, H of the problem: ρ t + divρu =, ρu t + div ρu u + aρ γ + δρ β = H H + µ u + λ + µu, 5.1 H t u H = ν H, divh =, satisfying the initial boundary conditions and 1.2. Moreover, ρ L β+1, T and the continuity equation in 5.1 holds in the sense of renormalized solutions. Furthermore, ρ, u, H satisfy the following uniform estimates: sup ρt γ L γ ce δ[ρ, m, H ], 5.2 t [,T ] δ sup ρt β L β ce δ[ρ, m, H ], 5.3 t [,T ] sup t [,T ] ρtut 2 L 2 ce δ[ρ, m, H ], 5.4 u L 2 [,T ];H 1 ce δ [ρ, m, H ], 5.5

11 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 11 where the constant c is independent of δ > and E δ [ρ, m, H ] = sup Ht 2 L 2 ce δ[ρ, m, H ], 5.6 t [,T ] H L 2 [,T ];H 1 ce δ [ρ, m, H ], m,δ 2 + a 2 ρ,δ γ 1 ργ,δ + δ β 1 ρβ,δ H 2 dx. Observing that the conditions imply that the term E δ [ρ, m, H ] appearing in can be indeed majored to be a constant E[ρ, m, H ] which is also independent of the choice of δ. This gives us a list of a priori estimates on ρ, u, H which are independent of δ. We omit the proof of Proposition 5.1, and concentrate our attention on passing to the limit in the artificial pressure term to establish the weak sequential stability property for the approximate solutions obtained in Proposition 5.1 as δ On the integrability of the density. We first derive an estimate of the density ρ δ uniform in δ > to make possible passing to the limit in the term δρ β δ as δ. The technique is similar to that in [1]. Noting that the function bρ = ln1 + ρ satisfies the condition 2.8, and ρ δ, u δ, H δ are the solution to 5.1 in the sense of renormalized solutions, we have ρδ ln1 + ρ δ t + div ln 1 + ρ δ u δ + ln 1 + ρ δ divu δ = ρ δ Now we introduce an auxiliary operator { B : f L p : which is a bounded linear operator, i.e., } f = [W 1,p ] 3 B[f] W 1,p cp f L p for any 1 < p < ; 5.9 and the function W = B[f] R 3 solves the problem divw = f in, W =. 5.1 Moreover, if f can be written in the form f = divg for some g L r, g n =, then for arbitrary 1 < r <. Define the functions: ϕ i = ψtb i [ln1 + ρ δ B[f] L r cr g L r 5.11 ] ln1 + ρ δ dx, ψ D, T, i = 1, 2, 3, where ln1 + ρ δ dx = 1 ln1 + ρ δdx is the average of ln1 + ρ δ over. By virtue of 5.2 and 5.8, we get Therefore, from 5.9, we have ln1 + ρ δ C[, T ]; L p for any finite p > 1. ϕ i C[, T ]; W 1,p for any finite p > 1. In particular, ϕ i C [, T ] by the Sobolev embedding theorem. Consequently, ϕ i can be used as test functions for the momentum balance equation in 5.1. After a little bit lengthy but straightforward computation, we obtain: T 7 ψδρ β δ + aργ δ ln1 + ρ δ dxdt = I j, 5.12 j=1

12 12 XIANPENG HU AND DEHUA WANG where T I 1 = ψ I 2 =λ + µ I 3 = I 4 = I 5 = I 6 = I 7 = T T T T T δρ β δ + aργ δ dx T ln1 + ρ δ dxdt, ψln1 + ρ δ divu δ dxdt, ψ t ρ δ u i δb i [ln1 + ρ δ ] ln1 + ρ δ dx dxdt, ψµ xj u i δ ρ δ u i δu j δ x j B i [ln1 + ρ δ [ ψρ δ u i δb i ln1 + ρ δ ρ δ divu δ 1 + ρ δ ln1 + ρ δ ρ δ 1 + ρ δ ψρ δ u i δb i [divln1 + ρ δ u δ ] dxdt, ] ln1 + ρ δ dx dxdt, divu δ dx ] dxdt, ] ψ H δ H δ B i [ ln1 + ρ δ ln1 + ρ δ dx dxdt. Now, we can estimate the integrals I 1 I 7 as follows. 1 First, we see that I 1 is bounded uniformly in δ, from 5.2, 5.3, and the following property: ln1 + t lim t t γ =. 2 As for the second term, we also have T I 2 ψln1 + ρ δ divu δ dxdt c, by the Hölder inequality, 5.5, 5.2, and the following property: ln t lim t t γ =, where and throughout the rest of the paper, c > denotes a generic constant. 3 Similarly, for the third term, we have T I 3 ψ tρ δ u i δb i [ln1 + ρ δ ln1 + ρ δ dx] dxdt c. Here, we have used 5.4, 5.5, and the embedding W 1,p L for p > 3, since ln1 + ρ δ ln1 + ρ δ dx L p for any 1 < p <. 4 Similarly to 3, we have T ] ψ xj u i δ xj B i [ln1 + ρ δ ln1 + ρ δ dx dxdt c, and, by 5.2, 5.5, and Hölder inequality, we have T ] ψρ δ u i δu j δ x j B i [ln1 + ρ δ ln1 + ρ δ dx dxdt c. Here, we used the restriction γ > 3 2. Therefore, we obtain I 4 c.

13 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 13 5 Next, by Hölder inequality and 5.9, we have, since w := I 5 c T ln1 + ρ δ ψ ρ δ 1 2 L γ ρ δ u δ L 2 B i [w] 2γ dt c, L γ 1 ρ δ 1 + ρ δ divu δ ln1 + ρ δ ρ δ 1 + ρ δ for some 1 < r < 2, and here we have used the estimates 5.2, Similarly to 5, using 5.2, 5.4, and we have T I 6 ψρ δ u i δb i [divln1 + ρ δ u δ ] dxdt c. Here, we have also used the property Finally, using Hölder inequality again, we have I 7 c T ψ H δ L 2 H δ L 2 dt c. Here we used the result ϕ i C[, T ], 5.6, and 5.7. Consequently, we have proved the following result: divu δ dx L r, Lemma 5.1. The solutions ρ δ of system 5.1 also satisfies the following estimate T ψδρ β δ + aργ δ ln1 + ρ δ dxdt c, where the constant c is independent of δ >. Remark 5.1. Lemma 5.1 yields T δρ β δ + aργ δ ln1 + ρ δ dxdt c, where c does not depend on δ. Using the similar method to Lemma 4.1 in [1], it can be shown cf. [21, 1, 9] that the optimal estimate for the density ρ δ is the following: T δρ β δ + aργ δ ρθ δ dxdt c, where the constant c is independent of δ >, and θ > is a constant. But as shown later, our estimate in Lemma 5.1 is enough for our purpose. Define the set J δ k = {x, t, T : ρ δ x, t k}, k >, δ, 1. From 5.2, there exists a constant s, such that, for all δ, 1 and k >, We have the following estimate: T δρ β δ dxdt meas{, T J δ k} s k. J δ k δρ β δ dxdt + T δk β meas{} + δ,t Jk δ T δρ β δ dxdt χ,t J δ k ρ β δ dxdt. 5.13

14 14 XIANPENG HU AND DEHUA WANG Then, by the Hölder inequality in Orlicz spaces cf. [1] and Lemma 5.1, we obtain T T δ χ,t J δ k ρ β δ dxdt δ χ,t J k δ L N max{1, Mρ β δ dxdt} δ N 1 k s 1 max{1, T 21 + ρ β δ ln1 + ρβ δ dxdt} 1 k δ N 1 max{1, 4ln2T meas{} + 4β s N 1 k s 1 max{δ, 4ln2δT meas{} + 4δβ T {ρ δ 1} T ρ β δ ln1 + ρ δ dxdt} ρ β δ ln1 + ρ δ dxdt}, 5.14 where L M, and L N are two Orlicz Spaces generated by two complementary N- functions Ms = 1 + sln1 + s s, Ns = e s 5.15 s 1, respectively. Due to Lemma 5.1, we know that, if δ < 1, T max{δ, 4ln2δT meas{} + 4δβ ρ β δ ln1 + ρ δ dxdt} c, for some c > which is independent of δ. Combining 5.13 with 5.14, we obtain the estimate T 1 δρ β k δ dxdt T δk β meas{} + c N 1, s where c does not depend on δ and k. Consequently T 1 lim sup δρ β k δ δ dxdt c N s The right-hand side of 5.16 tends to zero as k. Thus, we have T dxdt =, which yields lim δ δρ β δ δρ β δ in D, T Passing to the limit. The uniform estimates on ρ in Lemma 5.1, and Proposition 5.1 imply, as δ, ρ δ ρ in C[, T ]; L γ weak, 5.18 u δ u weakly in L 2 [, T ]; H 1, 5.19 and H δ H weakly* in L 2 [, T ]; H 1 L [, T ]; L 2, divh = in D, T ; and, from Lemma 5.1 and Proposition 2.1 in [9], we have, as δ, 5.2 ρ γ δ ργ weakly in L 1 [, T ]; L 1, 5.21 subject to a subsequence. By 5.19, 5.2 and the compactness of H 1 L 2, we obtain, u δ H δ u H in D, T, 5.22

15 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 15 and H δ H δ H H in D, T, 5.23 as δ. On the other hand, by virtue of the momentum balance in 5.1 and estimates , we have, as δ, Similarly, we have, as δ, ρ δ u δ ρu in C[, T ]; L 2γ γ+1 weak H δ H in C[, T ]; L 2 weak Thus, the limits ρ, ρu, H satisfy the initial conditions of 1.2 in the sense of distribution. Since γ > 3 2, 5.24 and 5.19 combined with the compactness of H1 L 2 imply, as δ, ρ δ u δ u δ ρu u in D, T. Consequently, letting δ in 5.1 and making use of , ρ, u, H satisfies t ρ + divρu =, 5.25 t ρu + divρu u µ u λ + µ divu + a ρ γ = H H, 5.26 H t u H = ν H, divh =, 5.27 in D, T. Therefore the only thing left to complete the proof of Theorem 2.1 is to show the strong convergence of ρ δ in L 1 or, equivalently, ρ γ = ρ γ. Since ρ δ, u δ is a renormalized solution of the continuity equation 5.1 in D R 3, T, we have T k ρ δ t + divt k ρ δ u δ + T kρ δ ρ δ T k ρ δ divu δ = in D R 3, T, 5.28 where T k is the cut-off functions defined as follows: z T k z = kt for z R, k = 1, 2,... k and T C R is concave and is chosen such that { z, z 1, T z = 2, z 3. Passing to the limit for δ +, we obtain where and t T k ρ + divt k ρu + T k ρρ T kρdivu = in D, T R 3, T kρ δ ρ δ T k ρ δ divu δ T k ρρ T kρdivu weakly in L 2, T, T k ρ δ T k ρ in C[, T ]; L p weak for all 1 p < The effective viscous flux. In this section, we discuss the effective viscous flux pρ λ + 2µdivu. Similarly to [21, 1, 9], we prove the following auxiliary result: Lemma 5.2. Let ρ δ, u δ be the sequence of approximation solutions obtained in Proposition 5.1. Then, T lim ψ φaρ γ δ λ + 2µdivu δt k ρ δ dxdt δ + T = ψ φaρ γ λ + 2µdivuT k ρ dxdt, for any ψ D, T and φ D.

16 16 XIANPENG HU AND DEHUA WANG Proof. As in [1, 9], we consider the operators A i [v] = 1 [ xi v], i = 1, 2, 3 where 1 stands for the inverse of the Laplace operator on R 3. To be more specific, A i can be expressed by their Fourier symbol [ ] A i [ ] = F 1 iξi ξ 2 F[ ], i = 1, 2, 3, with the following properties see [1]: A i v W 1,s cs, v L s R 3, 1 < s <, A i v L q cq, s, v L s R 3, q finite, provided 1 q 1 s 1 3, A i v L cs, v L s R 3, if s > 3. Next, we use the quantities ϕ i t, x = ψtφxa i [T k ρ δ ], ψ D, T, φ D, i = 1, 2, 3, as the test functions for the momentum balance equation in 5.1 to obtain, T T = ψφaρ γ δ + δρβ δ λ + 2µdivu δt k ρ δ dxdt + µ + T T ψ xi φλ + µdivu δ aρ γ δ δρβ δ A i[t k ρ δ ] dxdt T T ψu i δ T ψ xj φ xj u i δa i [T k ρ δ ] u i δ xj φ xj A i [T k ρ δ ] + u i δ xi φt k ρ δ dxdt φρ δ u i δ t ψa i [T k ρ δ ] + ψa i [T k ρ δ T kρ δ ρ δ divu δ ] dxdt ψρ δ u i δu j δ x j φa i [T k ρ δ ] dxdt T k ρ δ R i,j [ρ δ u j δ ] φρ δu j δ R i,j[t k ρ δ ] dxdt ψφ H δ H δ A[T k ρ δ ] dxdt, 5.29 where the operators R i,j = xj A i [v] and the summation convention is used to simplify notations. Analogously, we can repeat the above arguments for equation 5.26 and the test functions ϕ i t, x = ψtφxa i [T k ρ], i = 1, 2, 3,

17 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 17 to obtain T ψφaρ γ λ + 2µdivuT k ρ dxdt T = + µ + T T ψ xi φλ + µdivu aρ γ A i [T k ρ] dxdt T T T ψ xj φ xj u i A i [T k ρ] u i xj φ xj A i [T k ρ] + u i xi φt k ρ dxdt φρu i t ψa i [T k ρ] + ψa i [T k ρ T k ρρdivu] dxdt ψρu i u j xj φa i [T k ρ] dxdt ψu i T k ρr i,j [φρu j ] φρu j R i,j [T k ρ] dxdt ψφ H H A[T k ρ] dxdt. 5.3 Similarly to [1, 9], it can be shown that all the terms on the right-hand side of 5.29 converge to their counterparts in 5.3. Indeed, with the relations and the Sobolev embedding theorem in mind, it is easy to see that it is enough to show T ψu i δ T T k ρ δ R i,j [φρ δ u j δ ] φρ δu j δ R i,j[t k ρ δ ] dxdt ψu i T k ρr i,j [φρu j ] φρu j R i,j [T k ρ] dxdt, because the properties of A i and the weak convergence of u in L 2 [, T ]; H 1 imply A i T k ρ δ A i T k ρ in C, T, R i,j T k ρ δ R i,j T k ρ weakly in L p [, T ] for all 1 < p <, and A i [T k ρ δ T kρρdivu δ ] A i [T k ρ T k ρρdivu] weakly in L2 [, T ]; H 1. From Lemma 3.4 in [1], we have T k ρ δ R i,j [φρ δ u j δ ] φρ δu j δ R i,j[t k ρ δ ] T k ρr i,j [φρu j ] φρu j R i,j [T k ρ] weakly in L r, i, j = 1, 2, 3, for some r > 1. Hence, we complete the proof of Lemma The amplitude of oscillations. The main result of this subsection reads as follows, and is essentially taken from [1] cf. Lemma 4.3 in [1]: Lemma 5.3. There exists a constant c independent of k such that lim sup T k ρ δ T k ρ L γ+1,t c. δ +

18 18 XIANPENG HU AND DEHUA WANG Proof. By the convexity of functions t pt, t T k t, one has T lim sup δ + T ρ γ δ T kρ δ ρ γ T k ρ dxdt = lim sup ρ γ δ ργ T k ρ δ T k ρ dxdt On one hand, we have for all y z, and thus, δ + T + y γ z γ = lim sup δ + y z T ρ γ ρ γ T k ρ T k ρ dxdt ρ γ δ ργ T k ρ δ T k ρ dxdt. γs γ 1 ds γ y T k y T k z γ y z γ, z s z γ 1 ds = γy z γ, z γ y γ T k z T k y γ T k z T k y γ T k z T k y for all z, y. On the other hand, T lim sup δ + = lim sup δ + 2 sup δ T c lim sup δ + c = γ T k z T k y γ+1, divu δ T k ρ δ divut k ρ dxdt T k ρ δ T k ρ + T k ρ T k ρ divu δ dxdt divu δ L 2,T lim sup δ + T k ρ δ T k ρ L 2,T lim sup T k ρ δ T k ρ γ+1 L δ + γ+1,t. T k ρ δ T k ρ L 2,T The relations 5.31, 5.32 combined with Lemma 5.2 yield the desired conclusion The renormalized solutions. We now use Lemma 5.3 to prove the following crucial result: Lemma 5.4. The limit functions ρ, u solve 5.25 in the sense of renormalized solutions, i.e., t bρ + divbρu + b ρρ bρdivu =, 5.33 holds in D R 3, T for any b C 1 R satisfying b z = for all z R large enough, say, z M, where the constant M may depend on b. Proof. Regularizing 5.28, one gets t S m [T k ρ] + divs m [T k ρ]u + S m [T k ρρ T kρdivu] = r m, 5.34 where S m [v] = v m v are the standard smoothing operators and r m in L 2 [, T ]; L 2 R 3 for any fixed k see Lemma 2.3 in [2]. Now, we are allowed to multiply 5.34 by b S m [T k ρ]. Letting m, we obtain t b[t k ρ] + divb[t k ρ]u + b T k ρt k ρ bt k ρdivu = b T k ρ[t k ρρ T kρdivu] in D, T R

19 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 19 At this stage, the main idea is to let k in We have since and T k ρ ρ in L p, T for any 1 p < γ, as k, T k ρ ρ L p,t lim inf δ + T kρ δ ρ δ L p,t, T k ρ δ ρ δ p L p,t 2p k p γ ρ δ γ L γ,t ckp γ Thus 5.35 will imply 5.33 provided we show b T k ρ[t k ρρ T kρdivu] in L 1, T as k. To this end, let us denote then Q k,m = {t, x, T T k ρ M}, T b T k ρ[t k ρρ T kρdivu] dxdt sup b z T k ρρ T kρdivu dxdt z M Q K,M sup b z lim inf T kρ δ ρ δ T k ρ δ divu δ L z M δ + 1 Q k,m sup z M b z sup u δ L 2 [,T ];H 1 lim inf T kρ δ ρ δ T k ρ δ L δ δ + 2 Q k,m c lim inf δ + T kρ δ ρ δ T k ρ δ L 2 Q k,m. Now, by interpolation, one has T kρ δ ρ δ T k ρ δ 2 L 2 Q k,m T kρ γ 1 γ δ ρ δ T k ρ δ L 1,T T kρ γ γ δ ρ δ T k ρ δ L γ+1 Q k,m. Similarly to 5.36, we have and, using T k zz T kz, T kρ δ ρ δ T k ρ δ L 1,T ck 1 γ sup ρ δ γ L γ ck1 γ, δ 1 2 T kρ δ ρ δ T k ρ δ L γ+1 Q k,m T k ρ δ T k ρ L γ+1,t + T k ρ L γ+1 Q k,m T k ρ δ T k ρ L γ+1,t + T k ρ L γ+1 Q k,m + T k ρ T k ρ L γ+1,t T k ρ δ T k ρ L γ+1,t + Mc + T k ρ T k ρ L γ+1,t From Lemma 5.3 and 5.38, we obtain lim sup T kρ δ ρ δ T k ρ δ L γ+1 Q k,m 4c + 2Mc, δ + which, together with , completes the proof of Lemma 5.4.

20 2 XIANPENG HU AND DEHUA WANG 5.6. Strong convergence of the density. Now, we can complete the proof of Theorem 2.1. To this end, we introduce a sequence of functions L k C 1 R: { zlnz, z < k, L k z = zlnk + z z T k s k s ds, z k. 2 Noting that L k can be written as L k z = β k z + b k z, 5.39 where b k satisfies the conditions in Lemma 5.4, we can use the fact that ρ δ, u δ are renormalized solutions of 5.1 to deduce t L k ρ δ + divl k ρ δ u δ + T k ρ δ divu δ =. 5.4 Similarly, by 5.25 and Lemma 5.4, we have t L k ρ + divl k ρu + T k ρdivu =, 5.41 in D, T. By 5.4, we can assume, as δ, L k ρ δ L k ρ in C[, T ]; L γ weak. Taking the difference of 5.4 and 5.41 and integrating with respect to t, we get L k ρ δ L k ρφ dx t = L k ρ δ u δ L k ρu φ + T k ρdivu T k ρ δ divu δ φ dxdt, for any φ D. Passing to the limit for δ and making use of 5.42, one obtains L k ρ L k ρφ dx t = + lim δ + L k ρ L k ρu φ dxdt t T k ρdivu T k ρ δ divu δ φ dxdt, for any φ D. Since the velocity components u i, i = 1, 2, 3, belong to L 2 [, T ]; W 1,2, one has the following see Theorem 4.2 in [9]: u dist[x, ] L2 [, T ]; L 2. Let us consider a sequence of functions φ m D which approximate the characteristic function of such that φ m 1, φ m x = 1 for all x such that dist[x, ] 1 m, and φ m x 2m for all x Taking the sequence φ = φ m as the test functions in 5.43, making use of the boundary conditions in 1.2, and passing to the limit as m, one has t t L k ρ L k ρ dx = T k ρdivu dxdt lim T k ρ δ divu δ dxdt δ + We observe that the term L k ρ L k ρ is bounded by At this stage, the main idea is to let k in By 5.2, we can assume ρ ε lnρ ε ρlnρ weakly star in L [, T ]; L α for all 1 α < γ.

21 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 21 We also have since, by 5.2, L k ρ ρlnρ in L [, T ]; L α as k for all 1 α < γ, lim rk =, where k rk := meas{x, t, T ρ δx, t k}; and because L k z zlnz, repeating the similar procedure to 5.14, we have L k ρ ρlnρ L [,T ];L α sup lim inf L kρ δ ρ δ lnρ δ L t [,T ] δ + [,T ];L α 2qk sup sup max{1, Mρ α δ lnρ δ α dx} δ t [,T ] 2qk sup sup max{1, ρ α δ lnρ δ α ln1 + ρ α δ lnρ δ α dx} δ t [,T ] 2qk sup sup max{1, cαmeas{} + cα ρ α δ lnρ δ α+1 dx} δ t [,T ] {ρ δ e} 2qk sup sup max{1, cαmeas{} + cα, γ ρ γ δ dx} δ t [,T ] cqk, as k, where the function M is defined in 5.15, c is a constant independent of δ and 1 1 qk := χ [ρδ k] LN N 1. rk Similarly, we have and, by Lemma 5.3, L k ρ ρlnρ in L [, T ]; L α as k, for all 1 α < γ, T k ρ T k ρ in L α [, T ]; L α as k, for all 1 α < γ Finally, making use of Lemma 5.2 and the monotonicity of the pressure see 5.31, we obtain the following estimate on the right hand side of 5.45: t t T k ρdivu dxdt lim T k ρ δ divu δ dxdt δ + t 5.47 T k ρ T k ρdivu dxdt. From 5.46 and the Sobolev embedding theorem, we see that the right hand side of 5.47 tends to zero as k. Accordingly, one can pass to the limit for k in 5.45 to conclude ρlnρ ρlnρ x, t dx =, for t [, T ] a.e Because of the convexity of the function z zlnz, we have which, combining with 5.48, implies ρlnρ ρlnρ, a.e. in, T, ρlnρt = ρlnρt, for t [, T ] a.e Theorem 2.11 in [9], combined with 5.49, implies ρ ε ρ, a.e. in, T.

22 22 XIANPENG HU AND DEHUA WANG From the estimate 5.2 on ρ, together with Proposition 2.1 in [9], again we know, ρ δ ρ weakly in L 1, T, subject to a subsequence. By Theorem 2.1 in [9], we know that for any η >, there exists σ > such that for all δ >, ρ δ t, x dxdt < η, E for any measurable set E, T with meas{e} < σ. On the other hand, by virtue of Egorov s Theorem, for σ > given above, there exists a measurable set E σ, T such that meas{e σ } < σ, and ρ δ x, t ρx, t uniformly in, T E σ. Therefore, we have ρ δ ρ dxdt,t ρ δ ρ dxdt + E σ ρ δ ρ dxdt,t E σ 2η + T meas{} sup x,t E c σ ρ δ ρx, t, 5.5 which tends to zero if we first let δ +, and then let η +. The strong convergence of the sequence ρ δ in L 1, T follows from 5.5. The proof of Theorem 2.1 is completed. 6. Large-Time Behavior of Weak Solutions Our final goal in this paper is to study the large-time behavior of the finite energy weak solutions, whose existence is ensured by Theorem 2.1. First of all, from Theorem 2.1, we have { ess sup t> Et + µ Du 2 + λ + µdivu 2 + ν H 2} dxdt E. 6.1 Following the idea in [11], we consider a sequence ρ m x, t := ρx, t + m; u m x, t := ux, t + m; H m x, t := Hx, t + m, for all integer m, and t, 1, x. It is easy to see that 6.1 yields uniform bounds of ρ m L [, 1]; L γ, H m L [, 1]; L 2 ρm u m L [, 1]; L 2, ρ m u m L [, 1]; L 2γ γ+1, which are independent of m. Moreover, we have lim m 1 u m 2 L 2 + H m 2 L 2 dt =. 6.2 Hence, choosing a subsequence if necessary, we can assume that, as m, ρ m x, t ρ s weakly in L γ, 1; u m x, t u s weakly in L 2 [, 1]; H 1 ; H m x, t H s weakly in L 2 [, 1]; H 1.

23 GLOBAL SOLUTIONS OF MAGNETOHYDRODYNAMIC EQUATIONS 23 Furthermore, ρ s dx lim inf ρ m t dx CE. m Therefore, from the Poincaré inequality and 6.2, we know lim m 1 u m 2 L 2 dt =. This, combined with the compactness of H 1 L 2, implies Similarly, we know that u s =, a.e in, 1. H s =, a.e in, On the other hand, by Sobolev inequality, Hölder inequality, 6.1 and 6.2, we have 1 lim ρ m u m 2 3γ + ρ m u m 2 dt =. 6.4 m L γ+3 L 6γ γ+6 Since ρ, u are solutions to 1.1 in the sense of renormalized solutions, one has, in particular, ρ t + divρu = in D, T. 6.5 Then taking test functions ϕx, t = ψtφx in 6.5, where ψt D, 1, φ D, we have, using integrating by parts, 1 1 ρ m φ dx ψ tdt + ρ m u m φψ dxdt =. Letting m and using 6.4, we obtain 1 ρ s φ dx ψ tdt =. This implies that ρ s must be independent of t, by the arbitrariness of ψ. Now, following the procedure used in Section 5, or in Lemma 4.1 in [1], one can obtain ρ γ+θ m is bounded in L 1, 1, independently of m >, for some θ >. Consequently, one has ρ γ m ρ γ weakly in L 1, Therefore, passing to the limit in the momentum balance equation of 1.1 and using 6.2, 6.4, we get ρ γ = in D. 6.7 Now, we show that the convergence in 6.6 is indeed strong. To this end, similarly to [11], we consider { } 1 Gz = z α, < α < min 2γ, θ, θ + γ so that bz = Gz γ may be used in 2.6. Consider the vector functions [Gρ γ m,,, ] and [ρ γ m,,, ] of the time variable t and the spatial coordinates x. Using 2.6 and 6.2, 6.4, we get Div[Gρ γ m,,, ] is precompact in W 1,q1 loc, 1, 6.8 for some q 1 > 1 small enough. Similarly, making use of the momentum balance equation in 1.1, 6.2, and 6.4, we obtain Curl[ρ γ m,,, ] is precompact in W 1,q 2 loc, 1, 6.9

24 24 XIANPENG HU AND DEHUA WANG for some q 2 > 1, where Divf, f 1, f 2, f 3 := f t + Σ 3 i=1 xi f i, and Curlf, f 1, f 2, f 3 := i f j j f i, x := t, i, j =,..., 3. Meanwhile, we can assume Gρ γ m Gρ γ weakly in L p2, 1, 6.1 and Gρ γ mρ γ m Gρ γ ρ γ weakly in L r, 1, 6.11 with p 2 = 1 α, = 1 p 1 p 2 r < 1. Using the L p -version of the celebrated div-curl lemma see [26], we deduce that from Gρ γ ρ γ = Gρ γ ρ γ As G is strictly monotone, 6.12 implies Gρ γ = Gρ γ. Since L 1 α is uniformly convex, this yields strong convergence in 6.6. Therefore, we have ρ m ρ s strongly in L γ, 1. This, combined with 6.6 and 6.7, gives ρ γ s = in the sense of distributions, which implies that ρ s is independent of the spatial variables. Finally, by the energy inequality, the energy converges to a finite constant as t goes to infinity: E := lim Et, t and, by 6.4 and 6.3, m+1 lim ρ u 2 dx =, m m m+1 H 2 dx =. Thus E = lim = m lim m m+1 m a γ 1 ργ s dx. m 1 2 ρu2 + a γ 1 ργ H 2 dxdt Furthermore, using the continuity equation in 1.1, one easily observe that Thus, we have E = lim t ρx, t ρ s weakly in L γ as t. a γ 1 ργ s dx lim inf t 1 2 ρu2 + a γ 1 ργ H 2 a γ 1 ργ dx lim sup t a γ 1 ργ dx dx = lim t Et = E. This implies a a lim t γ 1 ργ dx = γ 1 ργ s dx, and 2.9 follows since the space L γ is uniformly convex. This completes the proof of Theorem 2.2.

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26 26 XIANPENG HU AND DEHUA WANG [25] D. Wang, Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J. Appl. Math , [26] Z. Yi, An L p theorem for compensated compactness, Proc. Royal Soc. Edinburgh A , Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 1526, USA. address: xih15@pitt.edu Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 1526, USA. address: dwang@math.pitt.edu