ON THE INCOMPRESSIBLE LIMITS FOR THE FULL MAGNETOHYDRODYNAMICS FLOWS

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1 ON THE INCOMPRESSIBLE LIMITS FOR THE FULL MAGNETOHYDRODYNAMICS FLOWS YOUNG-SAM KWON AND KONSTANTINA TRIVISA Abstract. In this article the incompressible limits of wea solutions to the governing equations for magnetohydrodynamics flows on both bounded unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stoes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains the derivation of uniform bounds. 1. Introduction Magnetohydrodynamic flows arise in science engineering in a variety of practical applications such as in plasma confinement, liquid-metal cooling of nuclear reactors, electromagnetic casting. Here we consider the viscous compressible magnetohydrodynamic flows t ϱ + div x ϱu =, 1.1 t ϱu + div x ϱu u + 1 Ma 2 xp = div x S + 1 Al 2 J H + 1 Fr 2 ϱ xf, 1.2 q t ϱs + div x ϱsu + div x = σ 1.3 ϑ d Ma 2 dt 2 ϱ u 2 + ϱe + Ma2 2Al 2 H 2 Ma 2 dx = Fr 2 ϱ xf u dx 1.4 t H u H + ν H =, divh =, 1.5 where u is the vector field, ϱ is the density, ϑ is the temperature, J the electronic current H is the magnetic field, Ma is Mach number, Fr is Froude number, Al is Alfven number. The electronic current satisfies Ampère s law whereas the Lorentz force is given by J = H, 1.6 J H = div x 1 µ H H 1 2µ H 2 I, Mathematics Subject Classification. 35B4, 35D5, 76N1, 35B45. Key words phrases. Low Mach number, compressible viscous fluid, Helmholtz function. The wor of Y.K. was supported by the research fund of Dong-A University. The wor of K.T. was supported in part by the National Science Foundation under the Grants DMS-87815, DMS PECASE DMS

2 2 YOUNG-SAM KWON AND KONSTANTINA TRIVISA with µ being a permeability constant of free space, which here is assumed to be µ = 1 for simplicity of the presentation. The electronic current J, the electric field E the magnetic field H are related through Ohm s law J = σe + u H. 1.8 The interaction described by the theory of magnetohydrodynamics, collective effects, is governed by the Faraday s law, t H + E =, div x H =. 1.9 Taing into consideration 1.8 we are able to write 1.9 in the following form t H + H u + ν H = 1.1 where ν = σ 1. Motivated by several recent studies devoted to the scale analysis as well as numerical experiments related to the proposed model see Klein et al. [13], our analysis is based on the following physically grounded assumptions: The viscous stress tensor S is determined through Newton s rheological law S = µ x u + x u 2 3 div xui + ηdiv x ui, 1.11 where µ >, η are respectively the shear bul viscosity coefficients. The heat flux q is given by Fourier s law q = κ x ϑ, 1.12 with the heat conductivity coefficient κ >. The entropy production satisfies σ 1 Ma 2 S : u q ϑ + ν Ma2 ϑ ϑ Al 2 H The dimensionless parameters Mach, Froude Alfven numbers see Klein et al. [13] are here expressed in terms of as follows With this scaling, the system read t ϱ + div x ϱu =, Ma =, Fr =, Al = t ϱu + div x ϱu u xp = div x S + 1 H H + 1 ϱ xf, q t ϱs + div x ϱsu + div x = σ, ϑ d 2 dt 2 ϱ u 2 + ϱe + 2 H 2 dx = ϱ x F u dx, t H u H + ν H =, divh =, 1.15 where the entropy production rate σ 1 2 S : u q ϑ + ν H 2. ϑ ϑ We first notice that the global-in-time existence solutions for system , supplemented with physically relevant constitutive relations, has been studied by several authors, Ducomet Feireisl [2], Huang, Wang [9]. Low Mach number problems have

3 LOW MACH NUMBER FOR MHD 3 been investigated by many authors, starting with the wor by Klainerman, Majda [11] for the Euler equations Lions, Masmoudi [14] for the isentropic Navier Stoes equations. Similar results in the spirit of the analysis presented by Lions, Masmoudi [14], are the recent progress by Novotny, Feiriesl [6, 7] for the full Navier-Stoes Fourier system, by Kuuca [12] for magnetohydrodynamic flows where results on aspects of convergence of system were discussed. In this article, we establish the low Mach number limit for the system for a new scaling accommodating both the case of bounded domains with Dirichlet boundary conditions unbounded domains. The outline of this article is as follows: In Section 2 we present the fundamental principles of thermodynamics the constitutive relations satisfied by various nonlinear quantities in the system. In Section 3 we present two initial-boundary-value problems introduce the notion of wea solutions for the two problems. In Section 4 we present the main results of the article on the low Mach number problems on bounded domains with Dirichlet boundary condition unbounded domains. In Section 5 we present the proof of the low Mach number problem for bounded domains. In Section 6 we give the rigorous proof of the low Mach number problem for the magnetohydrodynamic flows on unbounded domains in the spirit of Feireisl[4]. 2. Thermodynamics The physical properties of the magnetohydrodynamics flows are reflected through various constitutive relations which are expressed as typically non-linear functions relating the pressure p = pϱ, ϑ, the internal energy e = eϱ, ϑ, the specific entropy s = sϱ, ϑ to the macroscopic variables ϱ, u, ϑ Constitutive relations. According with the fundamental principles of thermodynamics, the specific internal energy e is related to the pressure p, the specific entropy s through Gibbs relation 1 ϑds = De + pd, 2.16 ϱ where D denotes the differential with respect to the state variables ϱ, ϑ. The total energy E is given by We consider the following equations of state. 1. The pressure p = pϱ, ϑ is here expressed as E = 1 2 u 2 + e H p = p F + p R, p R = a 3 ϑ4, a >, 2.18 where p R denotes the radiation pressure. Moreover, we shall assume that p F = p M + p E, where p M is the classical molecular pressure obeying Boyle s law, while p E is the pressure of electron gas constituent behaving lie a Fermi gas in the degenerate regime of high densities /or low temperatures see Chapters 1,15 in Eliezer et al [3]. Thus necessarily p F taes the form p F = ϑ 5 ϱ 2 P, 2.19 where P C 1 [, satisfies ϑ 3 2 P =, P z > for all z, 2.2

4 4 YOUNG-SAM KWON AND KONSTANTINA TRIVISA < c v 5 3 P z P zz z c v for all z >, P z lim z z 5 3 = p > Condition 2.21 reflects the fact that the specific heat at constant volume is strictly positive uniformly bounded. The reader may consult [5] for more details further discussion. 2. Similarly, we set e = e F + e R, with e R = a ϱ ϑ4, 2.22 where e F = e F ϱ, ϑ, p F ϱ, ϑ are interrelated through the following equation of state characteristic for mixtures of mono-atomic gases. 3. In accordance with 2.16, 2.22 we set where p F ϱ, ϑ = 2 3 ϱe F ϱ, ϑ 2.23 s = s F + s R, with s R = 4a 3ϱ ϑ3, s F = S S z = 3 2 ϱ ϑ 3 2, P z P zz z Transport coefficients. The viscosity coefficients µ η are assumed to be continuously differentiable functions of the temperature satisfying < µ1 + ϑ α µϑ, H µ1 + ϑ α, µ ϑ, H M, ηϑ, H η1 + ϑ α 2.26 < µ1 + ϑ νϑ, H µ1 + ϑ 3, 2.27 where Finally, we tae 1 α < κ = κ F ϑ, H + κ R ϱ, ϑ, H, 2.29 where κ F, κ R are continuously differentiable functions satisfying < κ F κ F ϑ, H κ F 1 + ϑ α for all ϑ >, 2.3 < κ R ϑ 3 κ R ϱ, ϑ, H κ R ϑ 3 for all ϱ, ϑ > Similarly to the above, the presence of the extra heat conductivity coefficient κ R is related to the effect of radiation see Oxenius [15]. 3. Variational formulation 3.1. An initial-boundary-value problem. Let R 3 be a bounded domain with the boundary of class C 2+ν, ν >. Generalization to domains with Lipschitz boundaries is possible via a suitable approximation procedure see Poul [16]. Accordingly, system has to be supplemented with a suitable set of boundary conditions in order to obtain, at least formally, a mathematically wellposed problem. The concept of the wea solutions introduced below requires the energy flux through the boundary to be zero. Therefore we suppose together with u n =, 3.32 q n = 3.33

5 LOW MACH NUMBER FOR MHD 5 where n sts for the outer normal vector. In addition, the impermeability condition 3.32 is supplemented either with the complete slip boundary condition or Sn n =, 3.34 u n =, where the latter, combined with 3.33, gives rise to the stard no-slip boundary conditions We also propose the boundary condition on H 3.2. Variational solutions. u = H = Definition 3.1. We say that a quantity {ϱ, u, ϑ, H} is an admissible variational wea solution of the full magnetohydrodynamic flows MHD 1.15 supplemented with the initial data {ϱ, u, ϑ, H } provided that the following hold. T T The density ϱ is a non-negative function, ϱ L, T ; L 5 3, the velocity field u L 2, T ; W 1,2 ; R 3, ϱ, u represent a renormalized solution of equation 1.1 on a time-space cylinder, T, that is, the integral identity ϱbϱ t ϕ + ϱbϱu x ϕ bϱdiv x uϕ dx dt = ϱ Bϱ ϕ, dx holds for any test function ϕ D[, T any b such that b L C[,, Bϱ = B1 + ϱ 1 bz z 2 dz The balance of momentum holds in distributional sense, namely ϱu t ϕ + ϱu u : x ϕ p div x ϕ + 1 [ H H] ϕ dx dt 3.38 = T S : x ϕ 1 ϱ xf ϕ dx dt ϱ u ϕ, dx for any test function ϕ D[, T ; D; R 3 satisfying ϕ n =, or, in addition, ϕ = if the no-slip boundary condition 3.35 is imposed. All quantities appearing in 3.38 are supposed to be at least integrable, S, p obey the constitutive relations 1.11, 2.19, respectively. In particular, the velocity field u must belong to a Sobolev space L p, T ; W 1,q ; R 3, therefore it is legitimate to require u to satisfy the boundary condition 3.35 as the case may be, in the sense of traces. The total energy of the system is constant of motion. Specifically, T 2 2 ϱ u 2 + ϱeϱ, ϑ + 2 H 2 t ϕ + ϱ x F uϕdxdt =, 3.39 where for all ϕ D[, T.

6 6 YOUNG-SAM KWON AND KONSTANTINA TRIVISA The integral inequality T ϱs t ϕ + ϱsu x ϕ + q xϕ dx dt ϑ T S : x u q xϑ + ν H 2 ϕdt ϑ ϑ ϱ sϱ, ϑ ϕ, dx 3.4 is satisfied for any test function ϕ D[, T, ϕ. Here the quantities S q are given through the constitutive equations Moreover, similarly to the above, all quantities must be at least integrable on, T. In particular, ϑ belongs to a Sobolev space L q, T ; W 1,q. In addition, we require ϑt, x to be positive for a.a. t, x, T. The Maxwell equation verifies T T H t ϕ dxdt + T u H ν H ϕ dx dt = H φdxdt = for all ϕ [D[, T ] 3, φ D[, T We now introduce the wea solutions of the target system 1.15 called the Oberbec- Boussinesq approximation supplemented by the Maxwell equation. Definition 3.2. A trio {u, Θ, H} is said to be a variational solution of the target system of MHD supplemented with the boundary conditions on the initial conditions u = or u n =, H =, x Θ n = 3.42 u, = u, Θ, = Θ, H, = H, 3.43 if the following conditions hold U L, T ; L 2 ; R 3 L 2, T ; W 1,2 ; R 3, T div x U = a.e. on, T, U n = in the sense of traces, the integral identity ϱu t ϕ + ϱu U : x ϕ µ[ x U + T x U] : x ϕ dxdt T = H H ϕdxdt ϱu ϕ, dx holds for any test function ϕ D, T ; R 3, div x ϕ = in, ϕ n =. 3.44

7 LOW MACH NUMBER FOR MHD 7 the temperature Θ L, T ; L 2 ; R 3 L 2, T ; W 1,2, r + ϱᾱ Θ 1 Θdx =, a.e. on, T, the integral identity T ϱ c p Θ t ϕ + Θu x ϕ κ x Θ x ϕ dxdt κ ϑᾱ T = x F x ϕdx dt c p ϱ c p Θ ϕ, dx holds for any test function ϕ D, T, Θ n =, Θ, = Θ,. H L 2, T ; W 1,2 ; R 3, the integral identity T H t ϕ H U + ν H ϕ dxdt = H ϕ, dx, is satisfied for any test function ϕ D, T ; R Main results We now introduce a geometric condition on which plays a crucial role in the study of propagation of the acoustic waves. Let us consider the following problem: φ φ = λφ in, = on, 4.47 n where φ is constant on. We call a solution of the problem 4.47 trivial if λ = φ is constant. We also define that verifies assumption H if all solutions of the problem 4.47 are trivial. Notice that Schiffer s conjecture shows that every satisfies H except the ball Feireisl, Novotny, Petzeltova [8] gives an example of domain which is trivial. In two dimensional space, it is proven that every bounded, simply connected open domain R 2 whose boundary id Lipschitz but not real analytic satisfies H Result on bounded domains. We first mention a result of incompressible limit problems on bounded domains with Dirichlet boundary conditions. Theorem 4.1. Let us assume Ma =, Fr =, Al = {ϱ, u, ϑ, H } be a family of variational solutions to MHD system in the sense of Definition 3.1with the same initial conditions given in Theorem 4.1 the boundary conditions defined in 3.33, 3.35, Let us assume that the pressure p, the specific internal energy e, the specific entropy s are functions of the state variables ϱ, ϑ satisfying Gibbs equation 2.16 supplemented with the structural hypotheses In addition, suppose that the transport coefficients µ, η, κ satisfy Assume the initial condition as follows. ϱ, = ϱ, = ϱ + ϱ 1,, u, = u,, ϑ, = ϑ + ϑ 1, 4.48 H, = H, = H 1, 4.49

8 8 YOUNG-SAM KWON AND KONSTANTINA TRIVISA where ϱ = 1 1 ϱ, dx, ϑ = ϑ, dx, 4.5 ϱ, ϱ 1, u, u 1, ϑ, ϑ 1, H1, H as tends to where we have used wealy convergence in L. Then, up to subsequence, ϱ ϱ in C[, T ]; L 1 L, T ; L 5 3, ϑ ϑ in L 2, T ; W 1,2, u U strongly in L 2, T ; W 1,2 ; R 3, H H strongly in L 2, T ; L 2 wealy in L 2, T ; W 1, ϱ 1 = ϱ ϱ ϱ 1 wealy in L, T ; L 5 3, ϑ 1 = ϑ ϑ ϑ 1 wealy in L 2, T ; W 1,2, where the velocity {u, Θ, H} with r = ϱ 1 1 ϑᾱ2 ϱ F, Θ = ϱ p F ϱ, ϑ c ϑ + ϑ 1 ϑᾱ F p c p 4.53 solves a wea solution of O-B system in the sense of Definition 3.2 with the boundary condition u = the initial data U = P[U ], Θ, := Θ = ϑ + c v ϑ 1 2 ϑ c v ϱ 1 c p 3 ϱ c, H, = H 4.54 p where the Helmholtz s projection P = I Q, Q = 1 div Result on unbounded domains. We next study incompressible limit problems for magnetohydrodynamic flows on unbounded domains. Consider an unbounded domains R 3 with a compact regular boundary a family of bounded domains { } > verifying:,, dist[x, ] as, 4.55 for any x. We consider a variation solution to NSF system in the sense of Definition 3.1 on. Then the second main result is the following. Theorem 4.2. Let R 3 be a unbounded domain with a compact boundary of class C 2+ν, ν > a family of bounded domains { } > satisfies Let us assume Ma =, Fr =, Al =, F = {ϱ, u, ϑ, H } be a family of variational solutions on, T to NSF system in the sense of Definition 3.1 on with the same initial data as given in Theorem 4.1 the boundary conditions 3.32, 3.33, on. Let us assume that all of hypotheses in Theorem 4.1 hold. Then we have the same convergence of {ϱ, u, ϑ, H } on any compact K as given in Theorem 4.1 such that the limits {U, Θ, H} of {u, ϑ ϑ, H } solves the

9 LOW MACH NUMBER FOR MHD 9 NSF system in the sense of Definition 3.2 with the boundary condition u n = the initial data Notice that we put F = to simplify our problem on the unbounded domains. 5. Low Mach number limit on bounded domains 5.1. Energy inequality uniform bounds. In this section we are going to derive some estimates on the sequence {ϱ, u, ϑ } >. We first deduce ϱ tdx = ϱ, ϱ 1 tdx = We now use the total energy balance 3.39 the entropy inequality 3.41 in order to derive the dissipation equality 1 2 ϱ u H 2 ϱ ϱ F tdx 1 H ϑ ϱ, ϑ + 2 H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ tdx + ϑ [ 2 σ [, t] ] = 1 2 ϱ, u, H, 2 ϱ, ϱ F dx H ϑ ϱ, ϑ + 2 H ϑϱ,, ϑ, ϱ, ϱ H ϑ ϱ, ϑ tdx for a.e. t, T, where the Helmholtz function H ϑ is defined by H ϑ ϱ, ϑ = ϱ eϱ, ϑ ϑs ϱ, ϑ We now introduce the set of essential values O ess, 2, { } O ess := ϱ, ϑ R 2 ϱ/2 < ϱ < 2 ϱ, ϑ < ϱ < 2 ϑ the residual set 5.59 O res :=, 2 O c ess. 5.6 We next define the essential set residual set of points t, x as follows M ess, T, M ess = {t, x, T ϱ t, x, ϑ t, x O ess M res =, T M ess c. Finally, each measurable function g can be decomposed as we set 5.61 g = [g] ess + [g] res 5.62 [g] ess = g1 M ess, [g] res = g1 M res = g [g] ess In order to exploit relation 5.57, we need to investigate the structural properties of the Helmholtz function H ϑ. More precisely, we show that the quantity H ϑ ϱ, ϑ H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ

10 1 YOUNG-SAM KWON AND KONSTANTINA TRIVISA is non-negative strictly coercive, attaining its global minimum zero at ϱ, ϑ. The structural properties of the Helmholtz function H ϑ follow as Lemma 5.1. Let H ϑϱ, ϑ be the Helmholtz function defined in 5.58 ϱ >, ϑ > be constants. Let O ess, O res be the sets of essential residual values in Then there exist c i = c i ϱ, ϑ, i = 1,..., 4, such that c 1 ϱ ϱ 2 + ϑ ϑ 2 H ϑ ϱ, ϑ H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ 5.64 c 1 ϱ ϱ 2 + ϑ ϑ 2 for all ϱ, ϑ O ess. H ϑ ϱ, ϑ H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ { H ϑ ϱ, ϑ inf r,θ Oess H ϑr, Θ r ϱ H ϑ ϱ, ϑ } = c 3 ϱ, ϑ > 5.65 for all ϱ, ϑ O res. for all ϱ, ϑ O res. H ϑ ϱ, ϑ H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ 5.66 c 4 ϱeϱ, ϑ + ϱ sϱ, ϑ The proof of Lemma 5.1 follows similar line of argument as the ones provided in [6]. By virtue of the inequality 5.64, one can easily deduce { ϱ ϱ } ess C 2 t 2 { + ϑ ϑ } L 2 ess t 2 L 2 H ϑ ϱ, ϑ H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ tdx 5.67 by virtue of 2.22, 2.21,2.24 together with 5.66, it follows that {ϱ } res t 5 3 L {ϑ } res t 4 L 4 + {ϱ sϱ, ϑ } res t L1 C 2 H ϑ ϱ, ϑ H ϑϱ, ϑ ϱ ϱ H ϑ ϱ, ϑ tdx 5.68 for a.e. t, T.

11 LOW MACH NUMBER FOR MHD 11 Using the initial condition 4.51 applied to 5.57 together with 5.67, 5.68 yields { } ess sup ϱ ϱ t,t t ess 2 2 C L 2 { } ϑ ess sup ϑ t,t t 2 2 C ess L 2 {ϱ } res t L C {ϑ } res t 4 L 4 2 C {ϱ sϱ, ϑ } res t L1 2 C. In addition, we get ess sup t,t ϱ u t L 2 ;R 3 C ess sup t,t H t L2 ;R 3 C ess sup t,t H t L 2 ;R 3 C x u + x u T 2 3 div xu I C L 2,T 5.7 x ϑ L 2,T 2 C x logϑ L2,T 2 C. As a direct consequence of 5.65, it follows that ess sup t,t M res[t] 2 C In accordance with the Maxwell equation , we obtain thus we derive {H } > bounded in L 2, T ; W 1,2 ; R { t H } > bounded in L p, T ; W 1,2 ; R for a certain p > 1. Using Aubin-Lions lemma applied to the Maxwell equation 3.41 together with 5.7, 5.72, 5.73, 5.75 implies H H strongly L 2, T ; L 2 ; R Thus we get the boundary condition H = on. Applying the following Korn-Poincare inequality together with estimates , we get {u n } n=1 bounded in L 2, T ; W 1,2 ; R Proposition 5.1. Let R 3 be a bounded Lipschitz domain. Assume that r is a non-negative function such that < C 1 rdx, r α dx C for a certain α > 1.

12 12 YOUNG-SAM KWON AND KONSTANTINA TRIVISA Then v W 1,p ω;r 3 Cp, C x 1, C 2 v + x v 2 3 div xv + Lp ;R 3 for any v W 1,p ; R 3, 1 < p <. r v dx 5.2. Convergence of continuity equation. We will use the uniform estimate 5.75 to deduce u U wealy in L 2, T ; W 1,2 ; R up to a subsequence of { > }. In accordance with 5.69, we have [ ϱ ϱ ] ϱ 1 wealy in L, T ; L 2, 5.77 ess [ ϱ ϱ ] in res L, T ; L 5 3, 5.78 whence [ ϱ ϱ ] ϱ 1 wealy in L, T ; L 5 3, 5.79 thus we obtain ϱ ϱ in L, T ; L 5 3, 5.8 so we can tae the limit of in the continuity equation 3.37 to get T U x ϕdx dt = for all ϕ C c, T div x u = a.e., T Convergence of entropy balance. In this section we will use the uniform estimates established in the previous section in order to chec the convergence in the equations 3.44, 3.45, 3.46 to identify the limit problem. To do this, we need the general result we can see the detail proof in [6]. Proposition 5.2. Let {ϱ } >, {ϑ } > be two sequences of non-negative measurable functions such that [ϱ 1 ] ess ϱ 1, [ϑ 1 ] ess ϑ 1, where the convergence means wealy limit in L, T ; L 2 as Suppose that ϱ 1 ess = ϱ ϱ, ϑ 1 sup t,t Let G C 1 O ess be a given function. Then, [Gϱ, ϑ ] ess G ϱ, ϑ where wealy in L, T ; L 2. Moreover, if G C 2 O ess, then [Gϱ, ϑ ] ess G ϱ, ϑ G ϱ, ϑ = ϑ ϑ. M rest 2 C. G ϱ, ϑ ϱ 1 + [ϱ 1 ] ess G ϱ, ϑ ϑ 2 C. G ϱ, ϑ ϑ 1, ϑ [ϑ 1 L ] ess,t ;L 1

13 LOW MACH NUMBER FOR MHD 13 To begin with, we will use the unform estimate 5.69 Proposition 5.2 to get [ ϑ ϑ ] ϑ 1 wealy in L, T ; L 2, 5.81 ess passing to a suitable subsequence thus the estimates together with Proposition yield [ ϑ ϑ ] in L, T ; W 1, res We now investigate the limit problem from the entropy balance equation 3.41 we rewrite it again in the form T sϱ, ϑ s ϱ, ϱ ϑ t ϕ + u x ϕ dt = T ϱ, ϑ, H ϑ sϱ,, ϑ, s ϱ, ϱ ϑ, ϕ, dx ϑ x x ϕ dt + 1 < σ, ϕ > [M,C][,T ] 5.83 for any test function ϕ D[, T. To identify the limit problem of 5.84, we first control the first part of the entropy balance We write ϱ sϱ, ϑ s ϱ, ϑ = [ϱ ] ess [sϱ, ϑ ] ess s ϱ, ϑ [ ϱ ] res [sϱ, ϑ ] ess s ϱ, ϑ [ ϱ sϱ, ϑ ] +. res Following Proposition , we can easily see [sϱ, ϑ ] ess s ϱ, [ϱ ] ϑ s ϱ, ϑ ess ϱ ϱ 1 s ϱ, ϑ + ϑ ϑ We next investigate the limit of second line in In accordance with the estimate 5.69, [ ϱ ] res [sϱ, ϑ ] ess s ϱ, ϑ in L, T ; L By virtue of the structural hypotheses 2.18, 2.21, we get ϱ sϱ, ϑ C 1 + ϱ logϱ + ϱ logϑ log ϑ + ϑ The estimates together with 5.57 yield { ϑ3 + ϱ logϱ } in L, T ; L 4 3, 5.88 which implies { ϑ3 + ϱ logϱ u } res res in L, T ; L p ; R 3, for a certain p > We also obtain from the Sobolev imbedding theorem that { ϱ logϑ log ϑ } in L 2, T ; L p ; R 3, for a certain p > 1, 5.9 res

14 14 YOUNG-SAM KWON AND KONSTANTINA TRIVISA {ϱ logϑ log ϑ } u in L 2, T ; L q ; R 3, for a certain p > res Finally, we apply 5.88, 5.89, 5.9, 5.91 to 5.87 in order to obtain { ϱ sϱ, ϑ } in res L2, T ; L p ; R 3, for a certain p > 1, 5.92 {ϱ sϱ, ϑ } u in L 2, T ; L q ; R 3, for a certain p > res Finally, we have to identify the wea limit D [sϱ, ϑ ] ess s ϱ, [ϱ ] ϑ ess u D wealy in L 2, T ; L 3 2 ; R 3 To this end, we need the Div-Curl Lemma. Let us set [ [sϱ, ϑ ] ess s ϱ, U = [ϱ ] ϑ [sϱ, ϑ ] ess s ϱ, ess, [ϱ ] ϑ ess u V = [u,,, ]. [ ϱ, ϑ ϑ ] x ϑ ]ess 5.94 Using the previous estimates, U, V meet all hypotheses of the Div-Curl lemma thus we obtain [sϱ, ϑ ] ess s ϱ, [ϱ ] ϑ s ϱ, ϑ ess u ϱ ϱ 1 s ϱ, ϑ + ϑ 1 U 5.95 ϑ wealy in L 2, T ; L 3 2 ; R 3. In conclusion, let us tae the limit in the entropy balance equation 5.84 then we obtain T s ϱ, ϑ ϱ ϱ 1 s ϱ, ϑ + ϑ 1 t ϕ + U x ϕ dx dt ϑ = ϱ s ϱ, ϑ T ϱ 1 + κ ϱ, ϑ x ϑ ϑ 1 x ϕdx dt s ϱ, ϑ ϑ ϑ 1 ϕ, dx where we have used 5.74 κ ϱ, ϑ, = κ ϱ, ϑ Convergence of moment equation. To begin with, using two estimates , we obtain ϱ U ϱu wealy in L 2, T ; L 3 23 ; R 3, 5.97 where we have used the Sobolev imbedding theorem W 1,2 L 6 it follows from 5.7 that Hence ϱ u ϱu wealy in L, T ; L 5 4 ; R ϱ u u ϱu U wealy in L, T ; L 3 23 ; R

15 LOW MACH NUMBER FOR MHD 15 Actually, we do not now ϱu U = ϱu U due to the oscillations of the gradient component of the velocity field we postpone this part to hle the oscillations of the gradient component in the next section. We now investigate the estimate of temperature. Following , we get {ϑ } > is bounded in L, T ; L 4 L 2, T ; L 6, 5.1 which implies that S µ ϑ x U + T x U wealy in L p, T 5.11 for a certain p > 1 thans to the estimate 5.76 together with the hypothesis of µ We also need to get the uniform estimates for the magnetic field {H } >. Notice that the estimate of H in 5.7 implies H H wealy in L 2, T ; W 1,2 ; R Indeed, in virtue of the estimates , one gets 1 H H = H H H H strongly in L p, T ; R for a certain p > 1. We are now able to identify the limit problem of the moment equation Let us tae the limit in the moment equation 3.38 we get T = T ϱu t ϕ + ϱu U : x ϕ dx dt for any test function where we have assumed µ ϑ x U + T x U : x ϕ ϱ 1 x F ϕ H H dx dt ϱu ϕdx ϕ C c [, T ] ; R 3, div x ϕ =, u, U wealy in L ; R 3. If we assume ϱu U = ϱu U, the initial condition follows as 5.14 U, = P[U ] We next need to hle the pressure. begin with writing By virtue of 2.19, 2.21, we get pϱ, ϑ = [pϱ, ϑ ] ess + [pϱ, ϑ ] res. t,t [ 1 ] C + res [pϱ, ϑ ] res which implies from 5.69, 5.71 that ess sup [pϱ, ϑ ] res [ ϱ 5 3 ] + res L1 [ ϑ 4 ], 5.16 res C. 5.17

16 16 YOUNG-SAM KWON AND KONSTANTINA TRIVISA From Proposition , multiplying the moment equation 3.38 by provides T p ϱ, ϑ ϱ 1 p ϱ, ϑ + ϑ 1 T div x ϕdx dt = ϱ x F ϕdx dt 5.18 ϑ for all ϕ C c, T ; R 3. Notice that we may assume F dx = by putting F = F F dx. Thus, the relation 5.18 yields ϱ 1 = ϑp ϱ p ϱ, ϑϑ 1 ϱ + F ρ p ϱ, ϑ Using 5.19 Gibbs equation, we obtain T ϱc p ϱ, ϑθ t ϕ + U x ϕ dx dt T κ ϱ, ϑ x Θ x ϕdx dt + κϑα T x F ϕdxdt ϑ c p = ϱ ϑc p ϱ, ϑ ϕ, dx, where we have here set ϑ 1 = Θ, c p ϱ, ϑ s ϱ, ϑ = ϱ 1 + c p ϱ, ϑθ, s ϱ, ϑ = ϱ 1 + s ϱ, ϑ ϑ 1 ϑ s ϱ, ϑ ϑ 1 + α ϱ, ϑ ϑf. Finally, from the relation 5.19, we get the Boussinesq relation where 5.11 r + ϱα ϱ, ϑθ =, r = ϱ 1 ϱ F. ρ p ϱ, ϑ 5.5. Convergence of the convective term. In general we do not expect ϱu U = ϱu U but our aim is to show that holds in the wea sense, namely, T T ϱu U : x ϕdx dt = [ ϱu U] : x ϕdx dt for any ϕ C c, T ; R 3, div x ϕ =. Before we prove 5.113, we will introduce the Helmholtz decomposition the following material may be found in most of the text boo of fluid mechanics.

17 LOW MACH NUMBER FOR MHD 17 Theorem 5.1. A vector function v : R 3 is written as where v = P[v] + Q[v], Q[v] = x Φ, Φ = div x v in, Notice that the Helmholtz projections v P[v], v Q[v] Φdx = map continously the spaces L p ; R 3 W 1,p ; R 3 into itself for any 1 < p. We now write ϱ u u = P[ϱ u ] u + Q[ϱ u ] P[u ] + Q[ϱ u ] Q[u ] In accordance with the uniform estimates , we obtain P[ϱ u ] P[ ϱu] = ϱu in C wea [, T ]; L 5 4 ; R In addition, we get ϱp[u ] u = P[ ϱ ϱ u ] + P[ϱ u ] u ϱ U 2 wealy in L due to the uniform estimates In particular, T T T P[u ] 2 dx dt = P[u ] u dx dt U 2 dx dt, which implies P[u ] U in L 2, T ; L Let us first investigate the estimate of P[ϱ u ] u in By virtue of 5.76 by employing the Sobolev imbedding theorem W 1,2 L 5, we deduce that P[ϱ u ] u ϱu U wealy in L 2, T ; L Moreover, following , we infer that P[ϱ u ] u wealy in L 2, T ; L In the previous discussion, it is sufficient to show T Q[ϱ u ] Q[u ] : x ϕdx dt, for any ϕ C c, T ; R 3, div x ϕ = in order to prove The Acoustic Waves. The acoustic equations are used to describe the time evolution of fast acoustic waves for compressible models. In the present context, we are dealing with the non-isentropic case, therefore we have to analyze further the issue of momentum oscillations of H [ϱ u ] because of the entropic temperature effects in our systems. To begin with, we write the pressure in the Taylor expansion., pϱ, ϑ p ϱ, ϑ = 4 ϑ4 ϑ 4 + p F ϱ, ϑ ϱ sϱ, ϑ s ϱ, ϑ = ϱ ϱ ϱ + p F ϱ, ϑ ϑ, ϑ ϑ + < D 2 p F a, bϱ ϱ, ϑ ϑ, ϱ ϱ, ϑ ϑ >, s ϱ, ϑ ϱ ϱ + ϱ s ϱ, ϑ ϑ ϑ ϑ + < D 2 ϱsϱ, ϑ s ϱ, ϑa, bϱ ϱ, ϑ ϑ, ϱ ϱ, ϑ ϑ >,

18 18 YOUNG-SAM KWON AND KONSTANTINA TRIVISA for certain a, b. In accordance with the Gibb s equation 2.16, one gets pϱ, ϑ p ϱ, ϑ = Λ 1 ϱ ϱ + Λ 2 sϱ, ϑ s ϱ, ϑ + Σa, b ϱ ϱ, ϑ ϑ, ϱ ϱ, ϑ ϑ, for a suitable Σ where p ϱ, ϑ Λ 1 = + 1 ϱ p ϱ, ϑ 2 s ϱ, ϑ 1, Λ2 2 = 1 ϱ p ϱ, ϑ s ϱ, ϑ 1 ϑ ϑ ϑ ϑ We now write the moment equation 3.37 the entropy balance 5.84 as follows. T V t ϕ + Λ 1 F x ϕ dx dt T = V, ϕ, dx + Π 1 x ϕdx dt + Λ 2 < σ, ϕ > for any ϕ D[, T T F t ϕ + V div x ϕ + Mdiv x x ϕ + T x ϕ 2 T T = F, ϕ, dx + Π 2 : x ϕdx dt 1 3 div xϕi F dx dt µ ϑ T x ϕ + T x ϕ + ϱ ϱdiv x 2 ϱ 2 3 div xϕi u dx dt + T Π 3 div x ϕ ϱ ϱ F ϕ [ H H ] ϕdx dt 5.125

19 LOW MACH NUMBER FOR MHD 19 for any test function ϕ D[, T ; R 3 where 2µ ϑ F = ϱ u, F, = ϱ, u,, M =, ϱ ϱ ϱ V = Λ 1 ϱ, ϱ V, = Λ 1 Π 1 = Λ 2 κϱ, ϑ ϑ sϱ, ϑ s ϱ, + Λ 2 ϱ ϑ, sϱ,, ϑ, s ϱ, + Λ 2 ϱ ϑ,, ϑ x sϱ, ϑ s ϱ, ϱ ϑ u Π 2 = µϑ µϑ x u + T x u 2 3 div xu I ϱ u u { Π 3 ϱ ϱ } = Λ 1 res + J { ϱ ϱ { ϱ sϱ, ϑ s ϱ, + Λ ϑ } 2 res, ϑ ϑ }, ess { ϱ ϱ, ϑ ϑ } ess { pϱ, ϑ p ϱ, ϑ } Integrations can be written as a variational formulation of wave equation: t V + Λ 1 div x F = g 1 res t F + x V Mdiv x [[ x F ]] = g 2 F, = F,, V, = V,, in, T where g 1 = div x Π 1 + Λ 2 σ g 2 = div x Π M x x ϱ u + x ϱ u T 2 3 div xϱ u + x Π ϱ ϱ + x F + H H [[M]] = M + MT tracemi. In accordance with the uniform estimates in the previous section, we obtain suitable estimates for g 1, g 2 to apply the same argument of the section 5-7 in Feireisl, Novotny [6]. Finally, we prove u u wealy in L 2, T ; W 1,2 ; R Analysis of eigenvalues of the acoustic operator. To begin with we study the spectrum of the following differential operator. [ ] [ ] [ ] v v v A + B q q q

20 2 YOUNG-SAM KWON AND KONSTANTINA TRIVISA where [ ] [ ] [ ] [ ] v divx q A = v, B = q Λ x v q Mdiv x [ x q ] In virtue of the no-slip condition, the operator B is to be supplemented with the boundary conditions q = while one taes q n = for A. We next study the eigenvalue problem [ ] [ ] [ ] v v v A + B = λ q q in, q q = 5.13 that can be viewed as a singular perturbation of [ ] [ ] v v A = λ q in, q n q = Notice that one easily chec that the problem admits a complete set of eigenvalues, namely, λ,, for which the corresponding eigenvalue [v, q ] tae the form v = constant,q L q σ; R 3 where L q σ; R 3 = {v D; R 3 div x v = in }, λ ±,n = ±i Λν n with theeigenfunctions [v n, q ±,n ], n = 1, 2,..., where ν n, v n are solutions of the Neumann eigenvalue problem v n = ν n v n in, x v n n =, ν n, q ±,n = 1 λ ±,n x v n, n = 1, 2, We now use the method of Vishi Ljusterni [18] in order to loo for approximate solutions of the perturbed problem 5.13 in the form of an asymptotic expansion in terms of. More precisely, [ v n q ±,n ] [ v n = q ±,n ] + λ ±,n [ v n,bl q ±,n,bl are to be found in order to solve [ ] [ ] v n A v n q ±,n + B q ±,n = λ ±,n + ] + = λ ±,n K =1 K [ =1 [ v n q ±,n λ ±,n v n,int q ±,n,int ] [ v n,bl + q ±,n,bl ] ] + r ±,n in, q ±,n = where the remainders r ±,n uniformly for. In 5.136, the quantities [v n,int x, q ±,n,int x] depends only on the spatial variables x while their boundary layer counter parts [v n,bl x, ξ, q ±,n,bl x, ξ] are functions of x the fast variables ξ = dx/, where d C 3 is a regularized boundary distance function, that means, dx > for x, dx = dist[x, ] for all x belonging to a sufficiently

21 LOW MACH NUMBER FOR MHD 21 small neighborhood of. The role of [v n,bl x, ξ, q ±,n,bl x, ξ] is to guarantee the satisfaction of the homogenous Dirichlet boundary conditions required in 5.13 that may be violated by the quantities [v n,int x, q ±,n,int x]. The following result represents one of the crucial ingredients of the proof of Theorem. Proposition 5.3. Let R 3 be a bounded domain of class C 3 such that the problem 4.47 admits only a trivial. Then for any K 1, there exists an L 2 orthogonal basis of eigenvectors {v n } n=1 solving functions v n,int C 2, q ±,n,int C 2 ; R 3, = 1,..., K v n,bl =, v n,bl W 2, C 2 [,, = 1,..., K q ±,n,bl W 2, C 2 [, ; R 3 such that the functions v n, q ±,n where 1 q <. Moreover, [ v n q ±,n r ±,n given by satisfy where L ;R 4 K Cq, n, ] [ ] v n q ±,n in L q ; R 4 as 5.14 Reλ ±,n 1 <, for any n = 1, 2, Strong convergence of velocity. In view of the previous result on the periodic domain, it only remains to prove that Q[u ] strongly in L 2, T ; L 2 ; R Q[ϱ u ] in L 2, T ; W 1,2 ; R In the previous section we have shown that {ϱ u } > are bounded in L, T ; L 5 4 ; R 4. Since the space L 5 4 ; R 4 is compactly imbedded into W 1,2 ; R 3, we can show { } t ϱ u t q ±,n dx in L 2, T for for any fixed q ±,n, n = 1, 2,... defined in Let us denote χ ±,n = V v n + ϱ u q ±,n dx where V is defined as follows ϱ ϱ V = Λ + 2 ϑ ϱ sϱ, ϑ 3 Following , we get ϱ u q ±,n dx = 1 2 χ±,n. χ,n. Consequently, in accordance with Proposition 5.3, it is enough to show where ω ±,n in L 2, T ω ±,n t = V tv n + ϱ u t q ±,n dx, t, T

22 22 YOUNG-SAM KWON AND KONSTANTINA TRIVISA [v n, q ±,n ] are the approximate eigenfunctions determined by We adopt test functions φ D in the acoustic equations 5.124, then we write T where ω ±,n t φ + λ ±,n T [ ω ±,n v φ dt + ϱ u T I 1 = Π 1 x v n φ dx dt, I 2 σ = Λ 2, vn φ, T I 3 = I 4 = 2µ ϑ ϱ T I 5 = Π 2 x q ±,n φdx dt, T T I 6 = ] r ±,n φdx dt = ϱ ϱdiv x [ x q ±,n ] u φdx dt, Π 3 div x q ±,n φdx dt, ϱ ϱ x F q ±,n φdx dt, T [ I 7 = H ] H q ±,n φdx dt, 6 I = with Π 1, Π 2, Π 3 are defined in To the proof end, we need to hle I, = 1,..., 6. Indeed, we get suitable estimates of I, = 1, 3, 4, 5, 6, with the same process in [6]. It remains to hle I 2, I 7. In virtue of the estimate 5.69, one gets I 2 = 2 < Γ, φ > [M;C][,T ] where {Γ } > is bounded in M + [, T ]. For I 7, we can use the estimate 5.7 for the magnetic field to obtain T I 7 = φtα tdt where {α } > is bounded in L p, T for a certain p > 1. Finally, we show ω ±,n in L τ, T as for any τ > with the same argument of Feireisl, Novotny s result [6]. Hence we can prove due to ω ±,n in L, T. We now chec the boundary conditions of the target system given in 3.2. Since u converges to U in L 2, T ; L 2 ; R 3, we gets U = on. In accordance with the estimates 5.81, 5.11, we conclude Sn n =, x Θ n = in the sense of trace which prove the boundary conditions given in 3.42.

23 LOW MACH NUMBER FOR MHD Low Mach number on the unbounded domains The only difference between Theorem 4.1 Theorem 4.2 is show for the gradient part H [V ] [ ] [ ] t V t, φdx t Vt, φdx in L 1, T for any φ Cc K; R 3, where V = ϱ u. In this section we will follow the framewor of Feireisl [4] based on a Kato s result [1], Theorem 6.1. Theorem 6.1. See Reed Simon [17] Let A be a closed densely defined linear operator H a self-adjoint densely defined operator in a Hilbert space M. For λ C R, let R H [λ] = H λid 1 denote the resolvent of H. Suppose that Then Γ := sup A R H [λ] A [v] X < λ C R,v DA, v =1 π sup A exp ith[w] 2 w X, w X =1 2 Xdt Γ We now introduce a time lifting Σ of the measure σ by where < Σ, ϕ >:=< σ, I[ϕ] >, I[ϕ]t, x := t ϕs, xds for any ϕ L 1, T ;. It is easy to see Σ L wea, T ; M+, where with < Σ τ, ϕ >:= lim δ + < σ, ψ δ ϕ >, for t [, τ, 1 ψ δ t = δ t τ for t τ, τ + δ, 1 for t τ + δ. Following the wave equation 5.127, we rewrite it in t X + Λ 1 div x F = div x G 1, t F + x X = div x G 2 + x G 3 + with the boundary condition V n = where X = V + Λ 2 2 Σ, F = ϱ u, G 1 = Π 1, G 2 = S ϱ u u, We write with H H + Λ 2 2 ω xσ G 3 = V pϱ, ϑ pϱ, ϑ 2. X := X 1 + X 2 X 1 = [V ] res + Λ 2 2 Σ, X 2 = [V ] ess

24 24 YOUNG-SAM KWON AND KONSTANTINA TRIVISA where in accordance with 5.69, 5.71, 5.85, ess sup t,t X 1 M C, ess sup We can also rewrite F into the following form: t,t F = F 1 + F 2, where in virtue of 5.69, 5.7 ess sup F 1 L1 ;R 3 C, ess t,t ess sup t,t sup t,t F 1 L 5/4 ;R 3 C Similarly, we get the uniform estimates in where T T X 2 L 2 C F 2 L2 ;R 3 C G 1 = G 1,1 + G 1,2, G 2 = G 2,1 + G 2,2 G 1,1 G 2,1 2 L 1 ;R 3 + G1,2 2 L 1 ;R 3 + G2,2 Finally, the remaining estimates follow as with T ess G 3 + Λ 2 2 ω = G3,1 sup t,t H 2 L 1 ;R 3 2 L 1 ;R dt C dt C G 3,1 M C H 2 L 1 ;R 3 We also use the uniform estimates 6.154, to obtain with the initial values dt C X C wea [, T ]; M, F C wea [, T ]; L 5/4 ; R 3, X, = X, M, F, = F, = ϱ, u, L 2 ; R 3. Moreover, it follows that where X, = X 1, + X 2,, X 1, M + X2, L 2 + F, L 2 ;R 3 C We will regularize X, V, G 1, G 2, G 3, H for variable x by a mollified function θ on in order to extend them on as discussed in Feireisl [4] we denote by X,δ, V,δ, G 1,δ, G2,δ, G3,δ, H,δ. For a fixed >, there exist smooth functions {X,,δ i } δ> Cc, i = 1, 2, {F i,,δ } δ> Cc, such that {X,,δ} 1,δ> is bounded in L 1, {X,,δ} 2,δ> is bounded in L {F,,δ },δ> is bounded in L 2 ; R 3,

25 LOW MACH NUMBER FOR MHD 25 X,,δϕdx 1 < X,, 1 ϕ >, X,,δϕdx 2 X,ϕdx 2 for any ϕ Cc, F,,δ ϕdx F, ϕdx for any ϕ Cc ; R 3, as δ tends to. Similarly, we get G 1,δ = G 1,1,δ + G1,2,δ, G1,i,δ C c, T ; R 3, i = 1, 2, G 2,δ = G 2,1,δ + G2,2,δ, G2,i,δ C c, T ; R 3 3, i = 1, 2, G 3,1,δ C c, T, [ H H ] θ δ C c, T ; R such that G 1,1,δ G1,1 in L 2, T ; L 1 ; R 3, G 1,2,δ G2,2 in L 2, T ; L 1 ; R 3 G 2,1,δ G2,2 in L 2, T ; L 1 ; R 3 3, G 2,2,δ G2,2 in L 2, T ; L 2 ; R 3 [ H H ] θ δ H H in L 2, T ; L 1 ; R 3 sup G 3,1,δ L 1 C, t [,T ] T G 3,1,δ ϕdxdt T < G 3,1, ϕ > dt for any ϕ Cc [, T ] as δ. In virtue of 6.152, we obtain the regularized initial value problem t X,δ + Λ 1 div x F,δ = div x G 1,δ, [ t F,δ + x X,δ = div x G 2,δ + x G 3,δ + H H ] θ δ + Λ 2 2 xσ,δ F,δ n = X,δ, = X,,δ, F,δ, = F,,δ. where means the convolution for variable x. System admits a finite speed of propagation of order Λ 1 / it can be seen by integrating the resulting expression over the set { Λ1 } t, x t [, τ], x, x < r t.

26 26 YOUNG-SAM KWON AND KONSTANTINA TRIVISA From the same argument of Feireisl [4], it is sufficient to show with the following initial boundary value problem t X + Λ 1 div x F = div x G 1 in, T, due to [ t F + x X = div x G 2 + H H ] in, T, ess F n =, X, = X,, F, = F, in, sup t,t F,δ F t, wdx as δ, together with the convergence of F,δ in where X, = X 1, + X 2,, X i, C c, i = 1, 2, F, C c ; R 3, G 1 = G 1,1 + G 1,2, G 1,i C c, T : R 3, i = 1, 2, G 2 = G 2,1 + G 2,2, G 2,i C c, T : R 3 3, i = 1, 2, H H C c, T ; R 3 with {X 1,} > is bounded in L 1, {X 2,} > is bounded in L 2 {F, } > is bounded in L 2 ; R 3 {G 1,1 } > is bounded in L 2, T ; L 1 ; R 3 {G 1,2 } > is bounded in L 2, T ; L 2 ; R {G 2,1 } > is bounded in L 2, T ; L 1 ; R 3 3 {G 2,2 } > is bounded in L 2, T ; L 2 ; R 3 3. Putting Λ 1 = 1 without loss of generality taing H to the initial boundary value problem with H [F ] = Φ, can be written by t X + Φ = div x G 1 in, T, t Φ + X = 1 div x [div x G 2 + H H ] in, T, with the boundary condition x Φ n = the initial conditions X, = X,, Φ, = Φ, = 1 [div x V, ] in. From the Duhamel s formula solving the

27 LOW MACH NUMBER FOR MHD 27 initial boundary value problem 6.169, we get Φ t = exp i t [ 1 2 Φ i ], + 2 [X,], + exp i t [ 1 2 Φ i ], 2 [X,], + + t t exp i t s exp i t s [ div xdiv x G 2 i + 2 [div xg 1 ] div x H [ 1 2 H ]ds 1 div xdiv x G 2 i 2 [div xg 1 ] div x H H ]ds In the sprite of Feireisl [4], We can prove for the gradient part H [V ] together with applying Theorem 6.1 with M = L 2, H =, A = ϕg, ϕ Cc, G Cc,. Indeed, x Φ wdx = Φ div x wdx = ϕφ div x wdx, = ϕg [Φ ]div x wdx + ϕg Id[Φ ]div x wdx, converges to as tends to if we show to apply the proofs of two terms of the last integrations in Feireisl [4] 1 div x H H ]ϕdx = Z ϕdx for any ϕ D D where Z L 2, T. Indeed, ϕ H H ] x Φdx is a bounded linear form for 1 ϕ D D where Φ = ϕ, the norm of which can be controlled by H H ] L1 ;R 3 1 thus we get on the Hilbert space D D together with using Riez representation Theorem the following observation 1 x Φ L 2 ;R 3 C x Φ W 1,2 ;R 3 C ϕ L 2 + L 2.

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