Incompressible limit of the non-isentropic ideal magnetohydrodynamic equations
aa r X i v : . [ m a t h . A P ] J a n INCOMPRESSIBLE LIMIT OF THE NON-ISENTROPIC IDEALMAGNETOHYDRODYNAMIC EQUATIONS
SONG JIANG, QIANGCHANG JU, AND FUCAI LI
Abstract.
We study the incompressible limit of the compressible non-isentropic ideal magnetohydrodynamic equations with general initial data inthe whole space R d ( d = 2 , Introduction
This paper is concerned with the incompressible limit to the compressible idealnon-isentropic magnetohydrodynamic (MHD) equations with general initial data inthe whole space R d ( d = 2 , ∂ t ρ + div( ρ u ) = 0 , (1.1) ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ p = ( ∇ × H ) × H , (1.2) ∂ t H − ∇ × ( u × H ) = 0 , div H = 0 , (1.3) ∂ t E + div ( u ( E ′ + p )) = div(( u × H ) × H ) . (1.4)Here ρ denotes the density, u ∈ R d the velocity, and H ∈ R d the magnetic field,respectively; E is the total energy given by E = E ′ + | H | / E ′ = ρ (cid:0) e + | u | / (cid:1) with e being the internal energy, ρ | u | / | H | / p = p ( ρ, θ ) and e = e ( ρ, θ ) relate the pressure p andthe internal energy e to the density ρ and the temperature θ of the flow.For smooth solutions to the system (1.1)–(1.4), we can rewrite the total energyequation (1.4) in the form of the internal energy. In fact, multiplying (1.2) by u and(1.3) by H respectively, and summing the resulting equations together, we obtain ddt (cid:16) ρ | u | + 12 | H | (cid:17) + 12 div (cid:0) ρ | u | u (cid:1) + ∇ p · u = ( ∇ × H ) × H · u + ∇ × ( u × H ) · H . (1.5)Using the identitiesdiv( H × ( ∇ × H )) = |∇ × H | − ∇ × ( ∇ × H ) · H , (1.6)div(( u × H ) × H ) = ( ∇ × H ) × H · u + ∇ × ( u × H ) · H , (1.7) Date : June 19, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Compressible ideal MHD equations, non-isentropic, incompressiblelimit. and subtracting (1.5) from (1.4), we thus obtain the internal energy equation ∂ t ( ρe ) + div( ρ u e ) + (div u ) p = 0 . (1.8)Using the Gibbs relation θ d S = d e + p d (cid:18) ρ (cid:19) , (1.9)we can further replace the equation (1.8) by ∂ t ( ρS ) + div( ρS u ) = 0 , (1.10)where S denotes the entropy.Now, as in [26], we reconsider the equations of state as functions of S and p ,i.e., ρ = R ( S, p ) for some positive smooth function R defined for all S and p > ∂R/∂p >
0. For instance, we have ρ = p /γ e − S/γ with γ > H = 0, thesystem (1.1)–(1.3) and (1.10) can be rewritten as A ( S, p )( ∂ t p + ( u · ∇ ) p ) + div u = 0 , (1.11) R ( S, p )( ∂ t u + ( u · ∇ ) u ) + ∇ p = ( ∇ × H ) × H , (1.12) ∂ t H − curl ( u × H ) = 0 , div H = 0 , (1.13) ∂ t S + ( u · ∇ ) S = 0 , (1.14)where A ( S, p ) = 1 R ( S, p ) ∂R ( S, p ) ∂p . Considering the physical explanation of the incompressible limit, we introducethe dimensionless parameter ǫ , the Mach number, and make the following changesof variables: p ( x, t ) = p ǫ ( x, ǫt ) = pe ǫq ǫ ( x,ǫt ) , S ( x, t ) = S ǫ ( x, ǫt ) , u ( x, t ) = ǫ u ǫ ( x, ǫt ) , H ( x, t ) = ǫ H ǫ ( x, ǫt ) , for some positive constant p . Under these changes of variables, the system (1.11)–(1.14) becomes a ( S ǫ , ǫq ǫ )( ∂ t q ǫ + ( u ǫ · ∇ ) q ǫ ) + 1 ǫ div u ǫ = 0 , (1.15) r ( S ǫ , ǫq ǫ )( ∂ t u ǫ + ( u ǫ · ∇ ) u ǫ ) + 1 ǫ ∇ q ǫ = (curl H ǫ ) × H ǫ , (1.16) ∂ t H ǫ − curl ( u ǫ × H ǫ ) = 0 , div H ǫ = 0 , (1.17) ∂ t S ǫ + ( u ǫ · ∇ ) S ǫ = 0 , (1.18)where we have used the abbreviations a ( S ǫ , ǫq ǫ ) := A ( S ǫ , pe ǫq ǫ ) pe ǫq ǫ = pe ǫq ǫ R ( S ǫ , pe ǫq ǫ ) · ∂R ( S ǫ , s ) ∂s (cid:12)(cid:12)(cid:12) s = pe ǫqǫ ,r ( S ǫ , ǫq ǫ ) := R ( S ǫ , pe ǫq ǫ ) pe ǫq ǫ . NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 3
Formally, we obtain from (1.15) and (1.16) that ∇ q ǫ → u ǫ → ǫ →
0. Applying the operator curl to (1.16), using the fact that curl ∇ = 0, andletting ǫ →
0, we obtain thatcurl (cid:0) r ( ¯ S, ∂ t v + v · ∇ v ) − (curl ¯ H ) × ¯ H ) = 0 , where we have assumed that ( S ǫ , q ǫ , u ǫ , H ǫ ) and r ( S ǫ , ǫq ǫ ) converge to ( ¯ S, , v , ¯ H )and r ( ¯ S,
0) in some sense, respectively. Finally, applying the identitycurl ( u × H ) = u (div H ) − H (div u ) + ( H · ∇ ) u − ( u · ∇ ) H , (1.19)we expect to get, as ǫ →
0, the following incompressible ideal non-isentropic MHDequations r ( ¯ S, ∂ t v + ( v · ∇ ) v ) − (curl ¯ H ) × ¯ H + ∇ π = 0 , (1.20) ∂ t ¯ H + ( v · ∇ ) ¯ H − ( ¯ H · ∇ ) v = 0 , (1.21) ∂ t ¯ S + ( v · ∇ ) ¯ S = 0 , (1.22)div v = 0 , div ¯ H = 0 (1.23)for some function π .The aim of this paper is to establish the above limit process rigorously in thewhole space R d .Before stating our main result, we briefly review the previous related works.We begin with the results for the Euler and Navier-Stokes equations. For well-prepared initial data, Schochet [32] obtained the convergence of the compressiblenon-isentropic Euler equations to the incompressible non-isentropic Euler equationsin a bounded domain for local smooth solutions. For general initial data, M´etivierand Schochet [26] proved rigorously the incompressible limit of the compressiblenon-isentropic Euler equations in the whole space R d . There are two key points inthe article [26]. First, they obtained the uniform estimates in Sobolev norms forthe acoustic component of the solutions, which are propagated by a wave equationwith unknown variable coefficients. Second, they proved that the local energy of theacoustic wave decays to zero in the whole space case. This approach was extendedto the non-isentropic Euler equations in an exterior domain and the full Navier-Stokes equations in the whole space by Alazard in [1] and [2], respectively, and tothe dispersive Navier-Stokes equations by Levermore, Sun and Trivisa [24]. For thespatially periodic case, M´etivier and Schochet [27] showed the incompressible limitof the one-dimensional non-isentropic Euler equations with general data. Comparedto the non-isentropic case, the treatment of the propagation of oscillations in theisentropic case is simpler and there is an extensive literature on this topic. Forexample, see Ukai [34], Asano [3], Desjardins and Grenier [7] in the whole space;Isozaki [14, 15] in an exterior domain; Iguchi [13] in the half space; Schochet [31]and Gallagher [10] in a periodic domain; and Lions and Masmoudi [29], and Des-jardins, et al. [8] in a bounded domain. Recently, Jiang and Ou [20] investigatedthe incompressible limit of the non-isentropic Navier-Stokes equations with zeroheat conductivity and well-prepared initial data in three-dimensional bounded do-mains. The justification of the incompressible limit of the non-isentropic Euler orNavier-Stokes equations with general initial data in a bounded domain or a multi-dimensional periodic domain is still open. The interested reader can refer to [5] onformal computations for the case of viscous polytropic gases and [4, 27] on some SONG JIANG, QIANGCHANG JU, AND FUCAI LI analysis for the non-isentropic Euler equations in a multi-dimensional periodic do-main. For more results on the incompressible limit of the Euler and Navier-Stokesequations, please see the monograph [9] and the survey articles [6, 30, 33].For the isentropic compressible MHD equations, the justification of the low Machlimit has been given in several aspects. In [21], Klainerman and Majda first studiedthe incompressible limit of the isentropic compressible ideal MHD equations in thespatially periodic case with well-prepared initial data. Recently, the incompressiblelimit of the isentropic viscous (including both viscosity and magnetic diffusivity)of compressible MHD equations with general data was studied in [12, 16, 17]. In[12], Hu and Wang obtained the convergence of weak solutions of the compressibleviscous MHD equations in bounded, spatially periodic domains and the whole space,respectively. In [16], the authors employed the modulated energy method to verifythe limit of weak solutions of the compressible MHD equations in the torus tothe strong solution of the incompressible viscous or partial viscous MHD equations(the shear viscosity coefficient is zero but the magnetic diffusion coefficient is apositive constant). In [17], the authors obtained the convergence of weak solutionsof the viscous compressible MHD equations to the strong solution of the idealincompressible MHD equations in the whole space by using the dispersion propertyof the wave equation if both shear viscosity and magnetic diffusion coefficients goto zero.For the full compressible MHD equations, the incompressible limit in the frame-work of the so-called variational solutions was studied in [22, 23, 28]. Recently,the authors [18] justified rigourously the low Mach number limit of classical solu-tions to the ideal or full compressible non-isentropic MHD equations with smallentropy or temperature variations. When the heat conductivity and large temper-ature variations are present, the low Mach number limit for the full compressiblenon-isentropic MHD equations was shown in [19]. We emphasize here that thearguments in [19] are completely different from the present paper (at least in thederivation of the uniform estimates), and depend essentially on the positivity offluid viscosities, magnetic diffusivity, and heat conductivity coefficients.As aforementioned, in this paper we want to establish rigorously the limit as ǫ → S ǫ , q ǫ , u ǫ , H ǫ ) | t =0 = ( S ǫ , q ǫ , u ǫ , H ǫ ) . (1.24)Our main result thus reads as follows. Theorem 1.1.
Suppose that the initial data S ǫ , q ǫ , u ǫ , H ǫ satisfy k ( S ǫ , q ǫ , u ǫ , H ǫ ) k H ( R d ) ≤ M (1.25) for some constant M > . Then there exist constants T > and ǫ ∈ (0 , such that for any ǫ ∈ (0 , ǫ ] , the Cauchy problem (1.15) – (1.18) , (1.24) has a unique NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 5 solution ( S ǫ , q ǫ , u ǫ , H ǫ ) ∈ C ([0 , T ] , H ( R d )) , and there exists a positive constant N , depending only on T , ǫ , and M , such that k (cid:0) S ǫ , q ǫ , u ǫ , H ǫ (cid:1) ( t ) k H ( R d ) ≤ N, ∀ t ∈ [0 , T ] . (1.26) Furthermore, suppose that the initial data ( S ǫ , u ǫ , H ǫ ) converge in H ( R d ) to ( S , w , H ) as ǫ → , and that there exist positive constants S , N and δ , such that S ǫ satisfies | S ǫ ( x ) − S | ≤ N | x | − − δ , |∇ S ǫ ( x ) | ≤ N | x | − − δ , (1.27) then the sequence of solutions ( S ǫ , q ǫ , u ǫ , H ǫ ) converges weakly in L ∞ (0 , T ; H ( R d )) and strongly in L (0 , T ; H s ′ loc ( R d )) for all s ′ < to a limit ( ¯ S, , v , ¯ H ) , where ( ¯ S, v , ¯ H ) is the unique solution in C ([0 , T ] , H ( R d )) of (1.20) – (1.23) with initialdata ( ¯ S, v , ¯ H ) | t =0 = ( S , w , H ) , where w ∈ H ( R d ) is determined by div w = 0 , curl ( r ( S , w ) = curl ( r ( S , v ) , r ( S ,
0) := lim ǫ → r ( S ǫ , . (1.28) The function π ∈ C ([0 , T ] × R d ) satisfies ∇ π ∈ C ([0 , T ] , H ( R d )) . We briefly describe the strategy of the proof. The proof of Theorem 1.1 in-cludes two main steps: uniform estimates of the solutions, and the convergencefrom the original scaling equations to the limiting ones. Once we have establishedthe uniform estimates (1.26) in Theorem 1.1, the convergence of solutions is easilyobtained by using the local energy decay theorem for fast waves in the whole spaceshown by M´etivier and Schochet in [26]. Thus, the main task in the present paperis to obtain the uniform estimates (1.26). For this purpose, we shall modify theapproach developed in [26]. In the process of deriving uniform estimates, when weperform the operator ( { E } − L ( ∂ x )) σ ( σ = 1 , , ,
4) to the continuity and momen-tum equations, or the operator curl to the momentum equations, one order higherspatial derivatives arise for the magnetic field, and the key point in the derivationof the uniform estimates is to cancel these troublesome magnetic terms in the fluidequations by the corresponding terms in the magnetic field equations. Only thosemagnetic terms of highest order derivatives are needed to be controlled (canceled).To achieve this cancellation, we restrict the Sobolev index s to be even, so that thehighest order derivatives applied to the momentum equations are not intertwinedwith the pressure equation, and we then apply the same highest order derivativeoperators to the magnetic field equations. We remark that the divergence-freeproperty of the magnetic field plays an important role in the process of the uniformestimates. Remark . In the proof of Theorem 1.1, it is crucial to appropriately choose theindex s of the Sobolev space H s to be even (here we take s = 4 for simplicity). Theresult is still valid for any even integer s > d/ H s for any integer s > d/ Remark . The uniform estimates obtained in this paper are still valid when weconsider our problem in a torus T d ( d = 2 ,
3) . However, it is an open problem toprove rigorously the incompressible limit of the compressible non-isentropic idealMHD equations in higher dimension T d due to the loss of dispersive estimatesfor the wave equations, see [27] for some discussions on the non-isentropic Eulerequations. SONG JIANG, QIANGCHANG JU, AND FUCAI LI
Remark . We point out that our arguments in this paper can be modified slightlyto the case of the non-isentropic MHD equations with nonnegative fluid viscositiesor magnetic diffusivity but without heat conductivity. In this case we do not requirethat the index s of the Sobolev space H s must be even. The reason is that we can usethe diffusion term (terms) to control the highest troublesome terms. For example,considering the situation that the magnetic diffusivity is zero and the fluid viscositycoefficients are positive, we can define M ǫ ( T ) := N ǫ ( T ) + Z T k u ǫ k s +1 d τ with N ǫ ( T ) := sup t ∈ [0 ,T ] k ( S ǫ , q ǫ , u ǫ , H ǫ )( t ) k s , s > d/ , and employ the same arguments with slight modifications to obtain the same resultas Theorem 2.1. Remark . It seems that the proof of the Theorem 1.1 can not be extended directlyto the full compressible MHD equations where the heat conductivity is positive. Inthis situation we can not write the total energy equation as a transport equationof entropy, which plays a key role in the proof. Instead, it is more convenient touse the temperature as an unknown in the total energy equation when the heatconductivity is positive, see [19] for more details.
Notions.
We denote by h· , ·i the standard inner product in L ( R d ) with norm k f k = h f, f i and by H k the usual Sobolev space W k, ( R d ) with norm k · k k .In particular, k · k = k · k . The notation k ( A , . . . , A k ) k means the summationof k A i k ( i = 1 , · · · , k ), and it also applies to other norms. For a multi-index α = ( α , . . . , α d ), we denote ∂ α = ∂ α x . . . ∂ α d x d and | α | = | α | + · · · + | α d | . We shallomit the spatial domain R d in integrals for convenience. We use the symbols K or C to denote generic positive constants, and C ( · ) and ˜ C ( · ) to denote smoothfunctions of their arguments which may vary from line to line.This paper is arranged as follows. In Section 2, we establish the uniform bound-eness of the solutions and prove the existence part of Theorem 1.1. In Section 3,we use the decay of the local energy to the acoustic wave equations to prove theconvergent part of Theorem 1.1.2. Uniform estimates
Throughout this section ǫ >
0, will be fixed, and the solution ( S ǫ , q ǫ , u ǫ , H ǫ ) willbe denoted by ( S, q, u , H ) and the corresponding superscript ǫ in other notationsare omitted for simplicity of presentation.In view of [26] and the classical local existence result in [25] for hyperbolicsystems, we see that the key point in the proof of the existence part of Theorem 1.1is to establish the uniform estimate (1.26), which can be deduced from the following a priori estimate. Theorem 2.1.
Let ( S, q, u , H ) ∈ C ([0 , T ] , H ( R d )) be a solution to (1.15) – (1.18) , (1.24) . Then there exist constants C > , < ǫ < and an increasing function C ( · ) from [0 , ∞ ) to [0 , ∞ ) , such that for all ǫ ∈ (0 , ǫ ] and t ∈ [0 , T ] , M ( T ) ≤ C + ( T + ǫ ) C ( M ( T )) , (2.1) NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 7 where M ( T ) := sup t ∈ [0 ,T ] k ( S, q, u , H )( t ) k . (2.2)The remainder of this section is devoted to establishing (2.1). In the calculationsthat follow, we always suppose that the assumptions in Theorem 1.1 hold. Weconsider a solution ( S, q, u , H ) ∈ C ([0 , T ] , H ( R d )) to the problem (1.15)–(1.18),(1.24) with the initial data satisfying (1.25).First, we have the following estimate of the entropy S , which was obtained in [26]. Lemma 2.2.
There exist a constant C > and a function C ( · ) , independent of ǫ , such that for all ǫ ∈ (0 , and t ∈ [0 , T ] , k S ( t ) k ≤ C + tC ( M ( T )) . (2.3)The following L -bound of ( q, u , H ) can be obtained directly using the energymethod due to the skew-symmetry of the singular term in the system and thespecial structure of coupling between the magnetic field and fluid velocity. This L -bound is very important in our arguments, since it is the starting point to thefollowing induction analysis used to get the desired Sobolev estimates. Lemma 2.3.
There exist a constant C > and an increasing function C ( · ) inde-pendent of ǫ , such that for all ǫ ∈ (0 , and t ∈ [0 , T ] , k ( q, u , H )( t ) k ≤ C + tC ( M ( T )) . (2.4) Proof.
Multiplying (1.15) by q , (1.16) by u , and (1.17) by H , respectively, integrat-ing over R d , and adding the resulting equations together, we obtain h a∂ t q, q i + h r∂ t u , u i + h ∂ t H , H i + h a ( u · ∇ ) q, q i + h r ( u · ∇ ) u , u i = Z [(curl H ) × H ] · u d x + Z curl ( u × H ) · H d x. (2.5)Here the singular terms involving 1 /ǫ are canceled. Using the identity (1.7) andintegrating by parts, we immediately obtain Z [(curl H ) × H ] · u d x + Z curl ( u × H ) · H d x = 0 . In view of the positivity and smoothness of a ( S, ǫq ) and r ( S, ǫq ), we get directlyfrom (1.15), (1.18) and the well-known nonlinear estimates (e.g., see [11]): k f ( u ) k σ ≤ C ( k u k σ ) k u k σ , u ∈ H σ ( R d ) , σ > d/ , f ( · ) smooth with f (0) = 0 , that k ∂ t S k ≤ C ( M ( T )) , k ǫ∂ t q k ≤ C ( M ( T )) , (2.6)while by the Sobolev embedding theorem, we find that k ( ∂ t a, ∂ t r ) k L ∞ ≤ k ( ∂ t a, ∂ t r ) k ≤ C ( M ( T )) . (2.7)By the definition of M ( T ) and the Sobolev embedding theorem, it is easy to seethat k ( ∇ a, ∇ r ) k L ∞ ≤ C ( M ( T )) . SONG JIANG, QIANGCHANG JU, AND FUCAI LI
Thus, from (2.5) we get that h aq, q i + h r u , u i + h H , H i ≤ (cid:8) h aq, q i + h r u , u i + h H , H i (cid:9)(cid:12)(cid:12) t =0 + C ( M ( T )) Z t (cid:8) | q ( τ ) | + | u ( τ ) | + | H ǫ ( τ ) | (cid:9) d τ. (2.8)Moreover, we have k q k + k u k ≤ k ( a ) − k L ∞ h aq, q i + k ( r ) − k L ∞ h r u , u i≤ C ( h aq, q i + h r u , u i ) , since a and r are uniformly bounded away from zero. Applying Gronwall’s Lemmato (2.8), we conclude that k ( q, u , H )( t ) k ≤ C k ( q , u , H ) k exp { tC ( M ( T )) } . Therefore, the estimate (2.4) follows from an elementary inequality e Ct ≤ Ct, ≤ t ≤ T , (2.9)where T is some fixed constant. (cid:3) To derive the desired higher order estimates, we shall adapt and modify thetechniques developed in [26]. Set E ( S, ǫq ) = (cid:18) a ( S, ǫq ) 00 r ( S, ǫq ) I d (cid:19) , L ( ∂ x ) = (cid:18) ∇ (cid:19) , U = (cid:18) q u (cid:19) , where I d denotes the d × d unit matrix.Let L E ( ∂ x ) = { E ( S, ǫq ) } − L ( ∂ x ) and r ( S ) = r ( S, r ( S ) issmooth, positive, and bounded away from zero with respect to each ǫ . First, usingLemma 2.2 and employing the same analysis as in [26], we have Lemma 2.4.
There exist constants C > , K > , and an increasing function C ( · ) , depending only on M , such that for all σ ∈ { , , , } , ǫ ∈ (0 , and t ∈ [0 , T ] , k U k σ ≤ K k L ( ∂ x ) U k σ − + ˜ C (cid:0) k curl ( r u ) k σ − + k U k σ − (cid:1) (2.10) and k U k σ ≤ ˜ C (cid:8) k{ L E ( ∂ x ) } σ U k + k curl ( r u ) k σ − + k U k σ − (cid:9) , (2.11) where ˜ C := C + tC ( M ( T )) + ǫC ( M ( T )) . We remark that the inequalities (2.10) and (2.11) are similar to the well knownHelmholtz decomposition, and the estimate on k S ( t ) k in Lemma 2.2 plays a keyrole in the proof of Lemma 2.4.Our next task is to bound k{ L E ( ∂ x ) } σ U k and k curl ( r u ) k σ − by inductionarguments. We first show the following estimate. Lemma 2.5.
There exists a constant C > and an increasing function C ( · ) from [0 , ∞ ) to [0 , ∞ ) , independent of ǫ , such that for all ≤ σ ≤ , all ǫ ∈ (0 , and t ∈ [0 , T ] , it holds that Z E |{ L E ( ∂ x ) } σ U ( t ) | d x ≤ C + tC ( M ( T )) , (2.12) NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 9 and for σ = 4 , Z E |{ L E ( ∂ x ) } σ U ( t ) | d x ≤ C + tC ( M ( T )) + 2 Z t Z ( ar ) − ( ∇ (∆ H ) H ) ∇ ∆div u d x d τ. (2.13) Proof.
Let U σ := { L E ( ∂ x ) } σ U , σ ∈ { , . . . , } . For simplicity, we set M := M ( T ),and E := E ( S, ǫq ). The case k = 0 is an immediate consequence of Lemma 2.3.It is easy to verify that the operator L E ( ∂ x ) is bounded from H k to H k − for k ∈ { , . . . , } . Note that the equations (1.15), (1.16) can be written as( ∂ t + u · ∇ ) U + 1 ǫ E − L ( ∂ x ) U = E − J (2.14)with J = (cid:18) H ) × H (cid:19) . For σ ≥
1, we commute the operator { L E } σ with (2.14) and multiply the result-ing system by E to infer that E ( ∂ t + u · ∇ ) U σ + 1 ǫ L ( ∂ x ) U σ = E ( f σ + g σ ) , (2.15)where f σ := [ ∂ t + u · ∇ , { L E } σ ] U , g σ := { L E } σ ( E − J ) . Multiplying (2.15) by U σ and integrating over (0 , t ) × R d with t ≤ T , noticingthat the singular terms cancel out since L ( ∂ x ) is skew-adjoint, we use the inequal-ities (2.6) and (2.7), and Cauchy-Schwarz’s inequality to deduce that12 h E ( t ) U σ ( t ) , U σ ( t ) i ≤ h E (0) U σ (0) , U σ (0) i + C ( M ) Z t k U σ ( τ ) k d τ + Z t k f σ ( τ ) k d τ + Z t Z E ( g σ U σ )( τ )d τ. (2.16)For σ ≤
3, the estimate (2.12) is obtained by taking a similar analysis to that ofLemma 2.4 in [26].Next we mainly deal with the case σ = 4. Following the proof process of Lemma2.4 in [26], we obtain that k f ( t ) k ≤ C ( M ( t )) . (2.17)Now we estimate the nonlinear term in (2.16) involving g σ . A straightforwardcomputation implies that U k = { L L } k − L u { L L } k − L q ! , if k is odd; (cid:18) { L L } k/ q { L L } k/ u (cid:19) , if k is even , where L := a − div , L := r − ∇ . Therefore for k = 4, we can replace the operator { L E ( ∂ x ) } by { L E ( ∂ x ) } = (cid:20) E − (cid:18) (cid:19) (cid:18) div 00 ∇ (cid:19)(cid:21) = (cid:18) ( ar ) − ∆
00 ( ar ) − ( ∇ div) (cid:19) + L , where L is a matrix whose elements are the summation of the lower order differ-ential operators up to order 3. Thus by virtue of Cauchy-Schwarz’s and Sobolev’sinequalities, and (2.7), we infer that Z t Z E g U d x d τ = Z t Z E ( L E ) ( E − J ) U d x d τ ≤ Z t Z ( ar ) − [( ∇ div) (curl H × H )]( ∇ div) u d x d τ + C ( M )=: I ( t ) + C ( M ) . To control I ( t ), we recall the basic vector identities:div( a × b ) = b · curl a − a · curl b , curl curl a = ∇ div a − ∆ a , and the fact that div H = 0 to deduce I ( t ) ≤ Z t Z ( ar ) − ( ∇ (∆ H ) H ) ∇ ∆div u d x d τ + C ( M ) . Thus, we have Z t Z E g U d x d τ ≤ Z t Z ( ar ) − ( ∇ (∆ H ) H ) ∇ ∆div u d x d τ + C ( M ) . (2.18)Finally, (2.13) follows from the above estimates (2.16)–(2.18) and the positivityof E . (cid:3) Next, we derive an estimate for k curl ( r u ) k σ − . Define f ( S, ǫq ) := 1 − r ( S ) r ( S, ǫq ) . (2.19)Hereafter we denote r ( t ) := r ( S ( t )) and f ( t ) := f ( S ( t ) , ǫq ( t )) for notationalsimplicity.One can factor out ǫq in f ( t ). In fact, using Taylor’s expansion, one obtains thatthere exists a smooth function g ( t ), such that f ( t ) = ǫg ( t ) := ǫg ( S ( t ) , ǫq ( t )) , k g ( t ) k ≤ C ( M ( T )) . (2.20)Since ∂ t S + ( u · ∇ ) S = 0 , the momentum equations (1.16) are equivalent to[ ∂ t + ( u · ∇ )]( r u ) + 1 ǫ ∇ q = g ∇ q + (1 − ǫg )(curl H ) × H . (2.21)We perform the operator curl to the system (2.21) to obtain that[ ∂ t + ( u · ∇ )](curl ( r u ))= [ u · ∇ , curl ]( r u ) + [curl , g ] ∇ q + curl [(1 − ǫg )(curl H ) × H ] , (2.22)and we can show NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 11
Lemma 2.6.
There exist constants C > , < ǫ < and an increasing function C ( · ) from [0 , ∞ ) to [0 , ∞ ) , such that for all ǫ ∈ (0 , ǫ ] and t ∈ [0 , T ] , it holds that k{ curl ( r u ) k ≤ C + tC ( M ( T )) , (2.23) and for | α | = 3 , X | α | =3 h ∂ α curl ( r u )( t ) , F ∂ α curl ( r u )( t ) i ≤ C + tC ( M ( T ))+ 2 X | α | =3 Z t Z ( ar ) − ( H · ∇ ) ∂ α H ∂ α (∆ u ) d x d τ, (2.24) where we denote by F := ( ar ) − ( r (1 − ǫg )) − > .Proof. The estimate (2.23) directly follows from an energy estimate performed on(2.22), and hence we omit the details here. Here we give a proof of (2.24) only.Set M := M ( T ), and ω = curl ( r u ). By the defination of f ( S, ǫq ) and (2.20),there exists an ǫ ∈ (0 , − ǫg is bounded from below for ǫ ∈ (0 , ǫ ].Taking ∂ αx ( | α | = 3) to (2.22), multiplying the resulting equations by F ∂ αx ω , andintegrating over (0 , t ) × R d with t ≤ T , we obtain that12 h ∂ α ω ( t ) , F ∂ α ω ( t ) i ≤ h ∂ α ω (0) , F ∂ α ω (0) i + C ( M ) Z t k ∂ α ω ( τ ) k d τ − Z t h ( u ǫ · ∇ ) ∂ α ω, F ∂ α ω i ( τ )d τ + Z t h [ u · ∇ , ∂ αx ] ω, F ∂ α ω i ( τ )d τ + Z t h ∂ α { [curl , g ǫ ] ∇ q } , F ∂ α ω i ( τ )d τ + Z t h ∂ α { [ u ǫ · ∇ , curl ]( r u ) } , F ∂ α ω i ( τ )d τ + Z t h ∂ α { curl [(1 − ǫg )(curl H ) × H ] } , F ∂ α ω i ( τ )d τ = : 12 h ∂ α ω (0) , F ∂ α ω (0) i + Z t X i =1 I i ( τ ) . (2.25)We have to estimate the terms I i ( τ ) (2 ≤ i ≤
6) on the right-hand side of (2.25).Applying integrations by parts, we have I ( τ ) ≤ Z R d F | ∂ α ω | div u d x + C ( M ) ≤ k div u ( τ ) k L ∞ k ∂ α ω ( τ ) k + C ( M ) ≤ C ( M ) , (2.26)while for the term I ( τ ), an application of Cauchy-Schwarz’s inequality gives | I ( τ ) | ≤ C k F ∂ α ω k k h α ( τ ) k , h α ( τ ) := [ u · ∇ , ∂ αx ] ω. The commutator h α is a sum of terms ∂ βx u ∂ γx ω with multi-indices β and γ satisfying | β | + | γ | ≤ | β | >
0, and | γ | >
0. Thus, k h α ( τ ) k ≤ C ( M ) , where the the following nonlinear Sobolev inequality has been used (see [11]): Forall α = ( α , · · · , α d ), σ ≥
0, and f, g ∈ H k + σ ( R d ), | α | = k , it holds that k [ f, ∂ α ] g k σ ≤ C ( k f k W , ∞ k g k σ + k − + k f k σ + k k g k L ∞ ) . Hence, we have | I ( τ ) | ≤ C ( M ) + k ∂ α ω ( τ ) k . (2.27)Noting that ([curl , g ] a ) i,j = a i ∂ x j g − a j ∂ x i g for a = ( a , · · · , a d ), using theestimate (2.20), and the inequality k uv k σ − k − l ≤ K k u k σ − k k v k σ − l for k ≥ l ≥ k + l ≤ σ and σ > d/ I ( τ ) as follows. | I ( τ ) | ≤ k ∂ α { [curl , g ] ∇ q }k k F ∂ α ω k≤ K k [curl , g ] ∇ q k k F ∂ α ω k≤ K k∇ g ( τ ) k k∇ q ( τ ) k k F ∂ α ω k≤ C ( M ) + k ∂ α ω ( τ ) k . (2.28)Similarly, the term I ( τ ) can be bounded as follows. | I ( τ ) | ≤ K k ∂ α { [ u · ∇ , curl ]( r u ) }k k F ∂ α ω k≤ K k [ u · ∇ , curl ]( r u ǫ ) k k F ∂ α ω k≤ K k [ u j , curl ] ∂ x j ( r u ) k k F ∂ α ω k≤ C ( M ) + k ∂ α ω ( τ ) k . (2.29)For the term I ( τ ), by virtue of the formula curl curl a = ∇ div a − ∆ a , weintegrate by parts to see that I ( τ ) = D ( ar ) − r (1 − ǫg ) ∂ α curl n (1 − ǫg )(( H · ∇ ) H − ∇ ( | H | o , ∂ α curl ( r u ) E ≤ − D ( ar ) − r (1 − ǫg ) ∂ α { (1 − ǫg ) H · ∇ ) H } , ∂ α curl curl ( r u ) E + C ( M ) ≤ −h ( ar ) − ∂ α [( H · ∇ )] H , ∂ α ( ∇ div u − ∆ u ) i + C ( M ) ≤ h ( ar ) − ( H · ∇ ) ∂ α H , ∂ α ∆ u i + C ( M ) , where we have used the fact that div H = 0.Inserting the estimates for I j ( τ ) (2 ≤ j ≤
6) into (2.25) and summing over | α | = 3, we obtain (2.24). The lemma is proved. (cid:3) We proceed to estimating the magnetic field.
Lemma 2.7.
There exist a constant C > and an increasing function C ( · ) from [0 , ∞ ) to [0 , ∞ ) , such that for | α | ≤ , all ǫ ∈ (0 , and t ∈ [0 , T ] , we have X | α |≤ Z ( ar ) − | ∂ α curl H | d x ≤ C + tC ( M ( T )) (2.30) and for | α | = 3 , X | α | =3 Z ( ar ) − | ∂ α curl H | d x ≤ C + tC ( M ( T )) − X | α | =3 Z t Z ( ar ) − ( H · ∇ ) ∂ α H ∂ α (∆ u ) d x d τ NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 13 − Z t Z ( ar ) − ( ∇ ∆ HH ) ∇ ∆div u d x d τ. (2.31) Proof.
Set M := M ( T ). We apply the operator curl to (1.17) and use the vectoridentity (1.19) to obtain that ∂ t (curl H ) + u · ∇ (curl H ) = − [curl , u ] · ∇ H + curl (( H · ∇ ) u − H div u ) . (2.32)Taking ∂ α ( | α | ≤
3) to (2.32), and multiplying the resulting equations by ( ar ) − × ∂ α curl H , and integrating over (0 , t ) × R d with t ≤ T , we find that h ( ar ) − ∂ α curl H , ∂ α curl H i = (cid:8)(cid:10) { a ( t ) r ( t ) } − ∂ α curl H ( t ) , ∂ α curl H ( t ) (cid:11)(cid:9)(cid:12)(cid:12) t =0 + h ∂ t { ( ar ) − } ∂ α curl H , ∂ α curl H i− Z t Z ( ar ) − ∂ α [( u · ∇ )curl H ) ∂ α curl H ] d x d τ − Z t Z ( ar ) − ∂ α { [curl , u ] · ∇ H } ( ∂ α curl H ) d x d τ + 2 Z t Z ( ar ) − ∂ α curl (( H · ∇ ) u − H div u )( ∂ α curl H ) d x d τ. For | α | ≤
2, it is easy to see that X | α |≤ Z ( ar ) − | ∂ α curl H | d x ≤ C + tC ( M ) . (2.33)Next, we treat the case | α | = 3. By a straightforward calculation we arrive at h ( ar ) − ∂ α curl H , ∂ α curl H i≤ C + tC ( M ) + 2 Z t Z ( ar ) − ∂ α curl (( H · ∇ ) u − H div u )( ∂ α curl H ) d x d τ =: C + tC ( M ) + 2 Z t I α ( τ )d τ. For the term I α ( τ ), using the fact that div H = 0, we have X | α | =3 I α ( τ ) = X | α | =3 Z ( ar ) − ∂ α curl (( H · ∇ ) u − H div u ) ∂ α curl H d x ≤ − X | α | =3 Z (cid:8) ( ar ) − ∂ α ( − ( u · ∇ ) H + ( H · ∇ ) u − H div u ) × ∂ α curl curl H (cid:9) d x + C ( M ) ≤ X | α | =3 Z ( ar ) − ∂ α (( H · ∇ ) u − H div u ) ∂ α ∆ H d x + C ( M ) ≤ − X | α | =3 Z ( ar ) − ( H · ∇ ) ∂ α H ∂ α (∆ u ) d x − Z ( ar ) − ( ∇ ∆ HH ) ∇ ∆div u d x + C ( M ) , where we have used the integration by parts several times in the last step. Thusthe lemma is proved, taking into account the positivity of a ( S, ǫq ) and r ( S, ǫq ). (cid:3) Finally, puting Lemmas (2.3)–(2.7) together, and using the induction argumentas in [26], we get the following estimate which finishes the proof of Theorem 2.1.
Lemma 2.8.
There exist a constant C > and an increasing function C ( · ) from [0 , ∞ ) to [0 , ∞ ) , such that for all ǫ ∈ (0 , ǫ ] and t ∈ [0 , T ] , k ( S, q, u , H )( t ) k s ≤ C + ( t + ǫ ) C ( M ( T )) . (2.34)3. Incompressible limit
In this section, we prove the convergence part of Theorem 1.1 by modifyingthe method developed by M´etivier and Schochet [26], see also some extensionsin [1, 2, 24].
Proof of the convergence part of Theorem 1.1.
The uniform bound (1.26) implies,after extracting a subsequence, the following limit:( q ǫ , u ǫ , H ǫ ) ⇀ ( q, v , ¯ H ) weakly- ∗ in L ∞ (0 , T ; H ( R d )) . (3.1)The equations (1.17) and (1.18) imply that ∂ t S ǫ and ∂ t H ǫ ∈ C ([0 , T ] , H ( R d )).Thus, after further extracting a subsequence, we obtain that, for all s ′ < S ǫ → ¯ S strongly in C ([0 , T ] , H s ′ loc ( R d )) , (3.2) H ǫ → ¯ H strongly in C ([0 , T ] , H s ′ loc ( R d )) , (3.3)where the limit ¯ H ∈ C ([0 , T ] , H s ′ loc ( R d )) ∩ L ∞ (0 , T ; H ( R d )). Similarly, by (2.22)and the uniform bound (1.26), we havecurl ( r ( S ǫ ) u ǫ ) → curl ( r ( ¯ S ) v ) strongly in C ([0 , T ] , H s ′ − ( R d )) (3.4)for all s ′ <
4, where r ( ¯ S ) = lim ǫ → r ( S ǫ ) := lim ǫ → r ( S ǫ , L (0 , T ; H s ′ loc ( R d )) for all s ′ <
4. To this end,we first show that q = 0 and div v = 0. In fact, from (2.14) we get ǫE ( S ǫ , ǫq ǫ ) ∂ t U ǫ + L ( ∂ x ) U ǫ = − ǫE ( S ǫ , ǫq ǫ ) u ǫ · ∇ U ǫ + ǫ J ǫ . (3.5)Since E ( S ǫ , ǫq ǫ ) − E ( S ǫ ,
0) = O ( ǫ ) , we have ǫE ( S ǫ , ∂ t U ǫ + L ( ∂ x ) U ǫ = ǫ h ǫ , (3.6)where, by virtue of (1.26), h ǫ is uniformly bounded in C ([0 , T ] , H ( R d )). Pass-ing to the weak limit in (3.6), we obtain ∇ q = 0 and div v = 0. Since q ∈ L ∞ (0 , T ; H ( R d )), we infer that q = 0. Now to complete the proof of Theorem1.1, we first prove the following proposition. Proposition 3.1.
Suppose that the assumptions in Theorem 1.1 hold, then q ǫ converges strongly to in L (0 , T ; H s ′ loc ( R d )) for all s ′ < , and div u ǫ convergesstrongly to in L (0 , T ; H s ′ − ( R d )) for all s ′ < . NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 15
The proof of Proposition 3.1 is built on the the following dispersive estimateson the wave equations established by M´etivier and Schochet [26] and reformulatedin [2].
Lemma 3.2 ( [2, 26]) . Let
T > and w ǫ be a bounded sequence in C ([0 , T ] ,H ( R d )) , such that ǫ ∂ t ( b ǫ ∂ t w ǫ ) − ∇ · ( c ǫ ∇ w ǫ ) = e ǫ , where e ǫ converges to strongly in L (0 , T ; L ( R d )) . Assume in addition that, forsome s > d/ , the coefficients ( b ǫ , c ǫ ) are uniformly bounded in C ([0 , T ] , H ( R d )) and converges in C ([0 , T ] , H ( R d )) to a limit ( b, c ) satisfying the decay estimates | b ( x, t ) − b | ≤ C | x | − − δ , |∇ x b ( x, t ) | ≤ C | x | − − δ , | c ( x, t ) − c | ≤ C | x | − − δ , |∇ x c ( x, t ) | ≤ C | x | − − δ , for some given positive constants b , c , C and δ . Then the sequence w ǫ convergesto in L (0 , T ; L ( R d )) .Proof of Proposition 3.1. We first show that q ǫ converges strongly to 0 in L (0 , T ; H s ′ loc ( R d )) for all s ′ <
4. An application of the operator ǫ ∂ t to (1.15) gives ǫ ∂ t ( a ( S ǫ , ǫq ǫ ) ∂ t q ǫ ) + ǫ∂ t div u ǫ = − ǫ ∂ t { a ( S ǫ , ǫq ǫ )( u ǫ · ∇ ) q ǫ } . (3.7)Dividing (1.16) by r ǫ ( S ǫ , ǫq ǫ ) and then taking the operator div to the resultingequations, one has ∂ t div u ǫ + 1 ǫ div (cid:16) r ( S ǫ , ǫq ǫ ) ∇ q ǫ (cid:17) = − div(( u ǫ · ∇ ) u ǫ ) + div (cid:16) r ( S ǫ , ǫq ǫ ) (curl H ǫ ) × H ǫ (cid:17) . (3.8)Subtracting (3.8) from (3.7), we get ǫ ∂ t ( a ( S ǫ , ǫq ǫ ) ∂ t q ǫ ) − div (cid:16) r ( S ǫ , ǫq ǫ ) ∇ q ǫ (cid:17) = F ( S ǫ , q ǫ , u ǫ , H ǫ ) , (3.9)where F ( S ǫ , q ǫ , u ǫ , H ǫ ) = ǫ div (cid:16) r ( S ǫ , ǫq ǫ ) (curl H ǫ ) × H ǫ (cid:17) − ǫ div(( u ǫ · ∇ ) u ǫ ) − ǫ ∂ t { a ( S ǫ , ǫq ǫ )( u ǫ · ∇ ) q ǫ } . In view of the uniform boundedness of ( S ǫ , q ǫ , u ǫ , H ǫ ), the smoothness and pos-itivity assumptions on a ( S ǫ , ǫq ǫ ) and r ( S ǫ , ǫq ǫ ), and the convergence of S ǫ , we findthat F ( S ǫ , q ǫ , u ǫ , H ǫ ) → L (0 , T ; L ( R d )) , and the coefficients in (3.9) satisfy the requirements in Lemma 3.2. Therefore, byvirtue of Lemma 3.2, q ǫ → L (0 , T ; L ( R d )) . On the other hand, the uniform boundedness of q ǫ in C ([0 , T ] , H ( R d )) and aninterpolation argument yield that q ǫ → L (0 , T ; H s ′ loc ( R d )) for all s ′ < . Similarly, we can obtain the convergence of div u ǫ . (cid:3) We continue our proof of Theorem 1.1. From Proposition 3.1, we know thatdiv u ǫ → div v in L (0 , T ; H s ′ − ( R d )) . Hence, from (3.4) it follows that u ǫ → v in L (0 , T ; H s ′ loc ( R d )) for all s ′ < . By (3.2), (3.3) and Proposition 3.2, we obtain r ǫ ( S ǫ , ǫq ǫ ) → r ( ¯ S ) in L ∞ (0 , T ; L ∞ ( R d )); ∇ u ǫ → ∇ v in L (0 , T ; H s ′ − ( R d )); ∇ H ǫ → ∇ ¯ H in L (0 , T ; H s ′ − ( R d )) . Passing to the limit in the equations for S ǫ and H ǫ , we see that the limits ¯ S and¯ H satisfy ∂ t ¯ S + ( v · ∇ ) ¯ S = 0 , ∂ t ¯ H + ( v · ∇ ) ¯ H − ( ¯ H · ∇ ) v = 0in the sense of distributions. Since r ( S ǫ , ǫq ǫ ) − r ( S ǫ ) = O ( ǫ ), we have( r ( S ǫ , ǫq ǫ ) − r ( S ǫ ))( ∂ t u ǫ + ( u ǫ · ∇ ) u ǫ ) → , whence, r ( S ǫ , ǫq ǫ )( ∂ t u ǫ + ( u ǫ · ∇ ) u ǫ ) = ( r ( S ǫ , ǫq ǫ ) − r ( S ǫ ))( ∂ t u ǫ + ( u ǫ · ∇ ) u ǫ )+ ∂ t ( r ( S ǫ ) u ǫ ) + ( u ǫ · ∇ )( r ( S ǫ ) u ǫ ) → r ( ¯ S )( ∂ t v + ( v · ∇ ) v )in the sense of distributions.Applying the operator curl to the momentum equations (1.16) and then takingto the limit, we conclude thatcurl (cid:0) r ( ¯ S )( ∂ t v + v · ∇ v ) − (curl ¯ H ) × ¯ H − µ ∆ v (cid:1) = 0 . Therefore, recalling curl ∇ = 0, we see that the limit ( ¯ S, v , ¯ H ) satisfies r ( ¯ S, ∂ t v + ( v · ∇ ) v ) − (curl ¯ H ) × ¯ H − µ ∆ v + ∇ π = 0 , (3.10) ∂ t ¯ H + ( v · ∇ ) ¯ H − ( ¯ H · ∇ ) v = 0 , (3.11) ∂ t ¯ S + ( v · ∇ ) ¯ S = 0 , (3.12)div v = 0 , div ¯ H = 0 (3.13)for some function π .If we employ the same arguments as in the proof of [26, Theorem 1.5], we findthat ( ¯ S, v , ¯ H ) satisfies the initial conditions (1.28). Moreover, the standard iterativemethod shows that the system (3.10)–(3.13) with initial data (1.28) has a uniquesolution ( S ∗ , v ∗ , H ∗ ) ∈ C ([0 , T ] , H ( R d )) . Thus, the uniqueness of solutions to thelimit system (3.10)–(3.13) implies that the above convergence results hold for thefull sequence ( S ǫ , q ǫ , u ǫ , H ǫ ). Thus, the proof is completed. (cid:3) Acknowledgements:
The authors are grateful to the anonymous referees for theirconstructive comments and helpful suggestions, which improved the earlier versionof this paper. The authors thank Professor Fanghua Lin for suggesting this problemand helpful discussions. This work was partially done when Li was visiting theInstitute of Applied Physics and Computational Mathematics in Beijing. He wouldlike to thank the institute for hospitality. Jiang was supported by the National
NCOMPRESSIBLE LIMIT OF COMPRESSIBLE NON-ISENTROPIC MHD EQUATIONS 17
Basic Research Program under the Grant 2011CB309705 and NSFC (Grant No.11229101). Ju was supported by NSFC (Grant No. 11171035). Li was supportedby NSFC (Grant No. 11271184, 10971094), PAPD, and the Fundamental ResearchFunds for the Central Universities.
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