Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain

This paper studies the convergence of the compressible isentropic magnetohydrodynamic equations to the corresponding incompressiblemagnetohydrodynamic equations with ill-prepared initial data in a periodic domain.We prove that the solution to the compressible isentropic magnetohydrodynamic equations with small Mach number exists uniformly in the time interval as long as that to the incompressible one does. Furthermore,we obtain the convergence result for the solutions filtered by the group of acoustics.

Here the unknowns ρ = ρ (t, x) ∈ R + denotes the dimensionless density of the fluid, u = u (t, x) ∈ R N the fluid velocity field, and H = H (t, x) ∈ R N the magnetic field, respectively. The spatial variable x ∈ T N a (N = 2, 3), where T N a denotes the torus with period 2πa i in the i th components, and a = (a 1 , . . . , a N ) with a i > 0(i = 1, . . . , N ). We assume that the (rescaled) pressure P = P (ρ ) is a smooth function of ρ . The constants µ > 0 and λ denote the shear and bulk viscosity coefficients of the flow, respectively. Physically, for the Newtonian flow, µ and λ satisfy the condition 2µ + N λ > 0.
In [21], cooperated with D. Wang, we proved that the compressible MHD equations (1)- (4) converge to the following incompressible MHD equations when tends to zero in the whole space R N with ill-prepared initial data: (v, B)| t=0 = (v 0 , B 0 ).
In the present paper, we study the same limit process in the perodic domain T N a with ill-prepared initial data. The main difference between the periodic case and the whole space case is that, for the whole space case, the oscillation is dispersive hence disappears when goes to zero, while for the periodic case, the oscillation will survive for ever. Thus, some new ideas must be introduced to deal with the oscillations. (For the well-prepared initial data case, there is no essitial difference between the whole space case and the periodic case, for example, see [22,18].) Without loss of generality, we assume that ρ ∼ 1 and P (1) = 1. We assume further that the initial data (ρ 0 = 1 + b 0 , u 0 , H 0 ) are uniformly bounded (in some functional space). Thus, we can set ρ := 1+ b and introduce the viscosity operator A := µ∆ + (λ + µ)∇div and the function I(z) := z 1+z , and denote p (1 + z) 1 + z = 1 + κz + zK(z) withK(0) = 0.
For presentation simplicity, we shall assume that the initial data (b 0 , u 0 , H 0 ) do not depend on and will be merely denoted by (b 0 , u 0 , H 0 ) satisfying v 0 = Pu 0 and B 0 = H 0 . Here P stands for the Leray projector on solenoidal vector fields and is defined by P := I − Q with Q := ∆ −1 ∇div. As for the general case, since Pu 0 tends to v 0 and H 0 tends to B 0 when goes to 0, it can be treated in a similar way.
Let us mention some previous mathematical works on the low Mach number limit to the isentropic MHD equations (1)-(3). Hu and Wang [16] proved the convergence of the weak solutions of the compressible MHD equations (1)-(3) to a weak solution of the viscous incompressible MHD equations (5)- (7) in the whole space, the torus, or the bounded domain. Jiang, Ju, and Li obtained the convergence of the weak solutions of the compressible MHD equations (1)- (3) to the strong solution of the ideal incompressible MHD equations (i.e. (5)- (7) with µ = ν = 0) in the whole space [19] or the viscous incompressible MHD equations in torus [17] with general initial data. Later, Feireisl, Novotny, and Sun [13] extended and improved the results in [19] to the unbounded domain case. Li [22] studied the invisid, incompressible limit of the viscous isentropic compressible MHD equations for local solutions with well-prepared initial data. Dou, Jiang, and Ju [11] studied the low Mach number limit for the compressible MHD equations in a 2D bounded domain with perfectly conducting boundary. Fan, Li, and Nakamura [12] studied the low Mach number limit for the compressible magnetohydrodynamic equations (1)-(3) in a 3D bounded domain with small initial data. It should be pointed out that all of the above results were carried out in the framework of Sobolev spaces.
As for the framework of Besov space, we first recall some related results on the isentropic Navier-Stokes system (namely, H = 0 in the equations (1)-(3)). In [5], Danchin proved the global well-posedness of isentropic Navier-Stokes equations in the critical Besov space when the initial data is a small perturbation around some given constant state. Later, the results of [5] were extended to more general Besov space in [2,4,15]. In [6,7,8], Danchin obtained the local well-posedness of solutions the isentropic Navier-Stokes equations with large initial data. In [9,10], Danchin studied the low Mach number limit of the isentropic Navier-Stokes equations in the whole space or the torus with ill-prepared data, respectively. In [24], the second author considered the low Mach number limit for the compressible MHD equations (1)-(3) in the whole space R N with small initial data in the critical Besov spaces. Cooperated with D. Wang [21], we established the convergence of compressible MHD equations to the incompressible MHD equations in whole space R N . Here we consider the low Mach number limit to the system (1)-(3) in the torus T N a with ill-prepared initial data.
As in [25], we introduce the following skew-symmetric operator L defined on It is easy to see that KerL is the set of couples (b, u) with b constant and div u = 0, and As pointed out in [10], the incompressible part of the velocity Pu isn't affected by the penalization. However, the compressible parts (b , Qu ) will experience high oscillations. In whole space R N , (b , Qu ) converge strongly to (0, 0) since the penalization operator has enough dispersive properties. However, in the torus T N a , we need to filter (b , Qu ) with the aid of the group L(τ ) := e −τ L generated by L. The filter method has been used by many authors in the study of incompressible limit, for example, see [25,14,23,17].
Denoting by L 1 (τ ) the first component of L(τ ), and by L 2 (τ ) the last N components. Letting η := λ + 2µ > 0 and defining V := L(− t ) T (b , Qu ). We then obtain the following equations of V : with Formally, using the non-stationary phase arguments, the equations (12) tends to the following limit system In the above system, v stands for the solution to the incompressible MHD equations (5)- (7), and the term Q i (i = 1, 2) is bilinear. Before stating our result, we introduce the following function spaces . For more details on the meanings of the above notations, see Appendix A blow. Now, we state our main result as follows. . Moreover, for a positive constant 0 depending only on δ, N, λ, µ, ν, p, b 0 , u 0 , H 0 , v and B, and for all 0 < ≤ 0 , the system (8)-(11) has a unique solution (b , u , H ) ∈ E N 2 +δ T0, η uniformly in satisfying that with δ < δ. Remark 1.1. When H = 0, the isentropic MHD equations (8)-(11) is reduced to the isentropic Navier-Stokes system. We point out that the results in Theorem 1.1 coincide with the ones obtained by Danchin [10] on the isentropic Navier-Stokes system. Thus, our results can be regarded as an extension of Danchin's results to the isentropic MHD equations.
To prove Theorem 1.1, besides the difficulties mentioned in [10], we encounter a few new difficulties. For instance, in order to obtain the priori estimates for the solution (see Proposition 4.1 below), we must find new appropriate reduced system analogous to the incompressible MHD equations (see (69)-(71) below). In this process, we need to do refined analyses to the nonlinear terms of the velocity and the magnetic field. Because of the strong coupling of the velocity and the magnetic field, the estimates for (69)-(71) are not as concise as the case of the isentropic Navier-Stokes equations. On the other hand, the coupling terms are analyzed in detail in each step of the proof of Theorem 1.1, see Sections 2-5 below. In our analysis, we shall make full use of the special structure of the isentropic compressible MHD equations.
Our paper is organized as follows. In Section 2, we prove that V tends to V . In Section 3, we study the convergence (Pu , H ) to (v, B). In Section 4, we obtain a priori estimates for (b , u , H ). In Section 5, based on the results obtained in Sections 2-4, we complete the proof of Theorem 1.1. Finally, we present an appendix to give the definition of the Littlewood-Paley decomposition, and recall some results in harmonic analysis.
Notation. Throughout the paper, the letter C stands for a generic constant independent of , and we use the notation A B as an equivalent to A ≤ CB. The notation A ≈ B means that both A B and B A hold simultaneously.
2. Convergence of the oscillating part of the system. Throughout the proof, below we shall use the following notations. Let s := N 2 + δ. For T ∈ (0, +∞], define: . Clearly, by means of Lemma A.3 and the definition of the space E s T,σ , the following inequality holds: In this section, we mainly conclude the following proposition. Proposition 2.1. Let T ∈ (0, +∞], (b , u , H ) ∈ E s T, η be a solution to the problem (8)- (11), and (v, B) ∈ F s T resp.V ∈ F s T be the solution to the problem (5)-(7) (resp. (13)- (14)). Then the following estimate holds: where C is a constant and τ tends to zero when goes to zero.
Since the proof of this proposition is similar to the case of the isentropic Navier-Stokes equations given by Proposition 3.1 in [10], we only give the sketch of the proof for brievity.
As in [10], we introduce the Hilbert basis (Φ α k ) α∈−1,1,k∈Z N \{0} of the space where sg(k) denotes a sign function on R N \{0} and its value is 1 if and only if the first nonzero component of k is positive, −1 elsewhere. Clearly, we have LΦ α k = −iαsg(k)|k|Φ α k . For any function A := α,kÂ α k Φ α k ∈ KerL ⊥ and L(τ ) = e −τ L , one obtains that Meanwhile, by letting v = kv k e ik·x , then Q 1 (u, B) and Q 2 (A, B) can be written as We define Q 1 (u, B) and Q 2 (A, B) as follows: When goes to zero, it is easy to check that Q i → Q i for i = 1, 2 in the sense of distributions. Finally, we have the convergence In the proof of Proposition 2.1, we need the following Lemma. For the heat equation of the following type we have withǨ,K ∈ L 2 (0, T ). Then there exists a universal constant c > 0 such that, for all t ∈ [0, T ], cµ .
Proof of Proposition 2.1. The equation for z reads with If we denote then R 1, and S can be rewritten as Then R 2, and R 3, read Then we need to make the change of function Further, we denote by R M "the low frequency part" of R := R 1, + R 2, + R 3, + S obtained by keeping only the indices (k, l) such that |k|, |l| ≤ M in the summations, and R ,M := R − R M . Clearly, one has According to (23) and (24), we obtain that Applying Lemma 2.1 to the equation (25), we gather that .
With regard to the estimates in (26), we shall use frequently the following facts: Firstly, we claim that R Clearly, the following inequalities hold Indeed, we denoteR 1, By the facts (27)-(29) and Lemma A.3, one has R 1, As forR 2, M ,R 3, M andS M , they can be treated similarly to that in [10]. Here, we only give their bounds as follows: S Define Collecting all the above estimates, then we conclude that (30) holds. It is easy to prove (31), we omit it here.
Next, we shall estimate the remaining terms on the right hand side of (26).
Estimates for R ,M : By (27)-(29) and Lemma A.3, one has We give the estimates for R 2, ,M , R 3, ,M and S ,M below and refer to [10] for more detail: Thus, we have Estimates for F : With the help of Lemma A.3, the following estimate holds:

FUCAI LI AND YANMIN MU
For some terms on the right hand side of (35), we have the following estimates: The other terms on the right hand side of (35) are as same as these in [10] and we just list their estimates as follows: v · ∇ω 1 L1 can be controlled by Estimates forR t, M : According to the expressions ofS t, . Using Lemma 2.2, we eventually gather that The remaining terms on the right hand side of (26) have the following bounds, and we can refer to [10] for more details: 3. Convergence of the incompressible part of the flow. This section is devoted to show the following proposition.
T be the solution to the problem (5)- (7). Then the following estimate holds on [0, T ]: where C > 0 is a positive constant.
First, according to the equations in (8)- (11) and (5)-(7), we deduce the following system for (ω 1 , ω 2 ): with A := Qu + v. Here we have used the fact that P(Qu · ∇Qu ) = 0. Before we give the proof of Proposition 3.1, we study the estimate of the heat system of the following type: We have Proof. For brevity, we first study the following simplified model: Localizing the system (48)-(49) by means of the Littlewood-Paley decomposition, and taking energy integration, finally, their summation, we get With regard to each term in the above equality, using Lemma 7.5 in [9], we have Therefore, we gather that Taking the time integration and using the Bernstein's inequality in the above inequality, we obtain that Multiplying the both sides of (50) by 2 q(s−1) , squaring and summing over q, and then extracting a root, finally we gather that Here we have used Minkowski's inequality to bound the last term.
. Adding the above inequalities into (51), we end up with Now, we return to the system (43)-(46). Let With the aids of Proposition A.4, it follows that Similarly,

FUCAI LI AND YANMIN MU
Therefore, (52) can be rewritten as By applying Gronwall's inequality, we complete the proof.
In order to show Proposition 3.1, we first make a brief analysis. Since Qu does not tend to zero in any spaceL r T (H σ ) as goes to zero, Qu ·v, Qu ·B, v ·∇Qu , B · ∇Qu are not likely to be small inL 1 T (H s−1 ). Therefore, we can't apply Proposition 41 to the system (41)-(42) directly. But we notice that V = L(− t ) ⊥ (b , Qu ) converge strongly to some limit V .
We make the following change of function as in Section 2: Therefore, one gets where R M is the low frequency part of R obtained by keeping only the indices l and k such that |k|, |l| ≤ M , with with initial data ψ M (0) = PR M (0). Applying Proposition 3.2 to the system (53)-(54) yields In order to simply the presentation of the estimates below, we first introduce some notations,Ṽ According to Lemma A.2 and the definitions of Sobolev spaces, we deduce that for any r ∈ [1, +∞], σ ∈ R, Estimates forR M : We claim According to Proposition A.2, and (56), we gather that . Therefore, we also have R

M,2 L∞
T (H s−1 ) ≤ CM P + P X . Similar arguments can be applied toR M,1 andR M (0), and we eventually arrive at which complete the proof of (58) and (59). Now we deal with the right-hand side of (55).
Estimates forR t, M : We claim that R t, As in [10], R t, Next, we deal with R t,

M,2 L1
T (H s−1 ) . We rewriteR t, M,2 as M . With the help of the spectral localization ofR t, M,2 and Remark A.2, we have . Collecting all the above estimates and combining with (61), we deduce that . On the other hand, using (10), it is straightforward to obtain . Finally, combining it with Lemma 2.2, we easily obtain (60).
Estimates forR ,M : We claim that As forR ,M

4.
A priori estimates for the solution. In this section, we mainly obtain a priori estimates of (b , u , H ) in the space E s T, η . Namely, we shall prove the following proposition.

FUCAI LI AND YANMIN MU
For the above system, we claim that and (c, v, B) be a solution to the problem (69)-(71). Denoteν = min{µ, η, ν} andν = η max{1, η/µ, η/ν}. Then the following estimate holds with a constant C depending only on N, p, r and s, HereṼ Proof. Since we need to estimate the hybrid norms, the following quantity k q must be controlled, If we apply standard energy method to the localized system (69)-(71), it is difficult to obtain suitable estimates for k q . For this purpose, we shall find some f q , which is equivalent to k q , namely, Clearly, according to (69)-(71), it is easy to check that the following five equalities hold: where (a|b) stands for the scalar product in L 2 (T N a ). To begin with, we deal with the case for q ≤ q 0 − 1. By means of (73), (75)-(77), it follows that 1 2 Noticing that 2 q η ≤ 2 and F(∆∆ q Qv) is supported in C(0, 5 6 2 q , 12 5 2 q ), we then obtain that For the term (∆ q T zj ∂ j v|∆ q v), we can rewrite it as With the help of Lemma 7.5 in [9], we arrive at Applying integration by parts to (S q−1 z j ∂ j ∆ q v|∆ q v), we have Similarly, it holds For the term (∆ q ∂ j T zj c|∆ q c), it can be rewriten as According to the definition of the paraproduct and the facts (109), the equality ∆ q T div z c = |q−q |≤3 ∆ q (S q −1 div z∆ q c) holds. It is straightforward to obtain that Therefore, it is easy to get Finally, in order to estimate the term η(∇∆ q T zj ∂ j c|∆ q v) + η(∆ q T zj ∂ j v|∇∆ q c), we can rewrite it as Clearly, we have And for η(S q−1 z j ∂ j ∆ q ∇c|∆ q v) + η(S q−1 z j ∂ j ∆ q v|∇∆ q c), applying integration by parts, we get Now, plugging (79)-(86) into (78), we arrive at Since f q ≈ k q , the time integration of (87) gives Next, we deal with the case: q ≥ q 0 . From (74)-(77), it follows that Noticing that 2 q η ≥ 2 and that F(∆ q v) is supported away from B(0, 5 3 ). Thus, we have As for the terms on the right hand side of (89), their estimates are similar to the corresponding terms on the right hand side of (78), thus we omit the details and give only their bounds as follows: Plugging (90)-(94) into (89), we gather that As f q ≈ k q , the time integration of (95) gives Here we have used the following inequality: Multiplying (88) by 2 q(s−1) , squaring it and summing over q ≤ q 0 − 1, likewise, multiplying (96) by 2 q(s−1) , squaring it and summing over q ≥ q 0 , summarizing the obtained inequalities and taking the square root of it, we eventually arrive at Next, we treat (v HF , B HF ). In fact, from (70) and (71), we obtain that Integrating in time for the above two inequalities and applying the previous estimates, we get, for a universal constant κ, Finally, we deduce the general case from (102). In fact, we take the following change of function, Clearly, if (b , u , H ) solves the problem (8)- (11), then (b,ũ,H) must be a solution to the problem (8) -(11) for = 1. Using Proposition A.1, straightforward computations yield the desired inequality for (b , u , H ).

5.
Proof of Theorem 1.1. In this section, we mainly prove Theorem 1.1. we shall follow and adopt some ideas developed in [10]. Briefly, we shall divided the proof into six steps: Step 1. The estimate of V − V in the space G Step 5. Global existence for the limit system (13)- (14).
Step 6. We complete the proof of Theorem 1.1 with the help of continuation argument.
The above statement may be obtained by going along the lines of the proof of Theorem 1.1 and Remark 2.2 in [21]. However, it has been assumed that = 1, b 0 ∈ . Using appropriate change of varible (see Section 4 ), we can get the case > 0 from = 1. Moreover, Theorem 1.1 and Remark 2.2 in [21] still hold in the Sobolev spaces setting. Collecting the above argument and Proposition 5.1, one can prove that the solution (b , u , H ) may be continued until the time T 0 or on R + if T 0 = +∞ when is suitably small. The reader can find more details in [21]. Therefore, (H) is fulfilled on [0, T 0 ] and meanwhile, Proposition , we can conclude that V tends to V in G Appendix A. Some results in harmonic analysis. Let us first recall the definition and some basic properties of Sobolev space. Most of the materials stated below can be found in [10]. We shall use the letter C to denote the positive constant which may change from line to line.
We shall denote by Fz = (ẑ k ) k∈Z N the Fourier series of a distribution z ∈ S (T N ) such that whereZ N := Z/a 1 × · · · × Z/a N and |T N | stands for the measure of T N . We first introduce a C ∞ symmetric function ϕ of one variable supported in {r ∈ R, 5/6 ≤ |r| ≤ 12/5} such that q∈Z ϕ(2 −q r) = 1 for r = 0.
We then define the following dyadic blocks: and the following low-frequency cut-off: Therefore, we have the following simple propositions: Next, using the Littlewood-Paley decomposition, we define the Sobolev and Besov space and their corresponding Hybrid spaces as follows: We shall use the notation u = u − |T N | −1 T N udx for any u ∈ S (T N ), and H s , B s stand for the subset of distribution u of H s , B s with zero average (i.e.û 0 = 0). Then, we state several propositions as follows: Proposition A.1 ([10]). The following properties hold: (1) Sobolev embeddings: (a) For all s ∈ R, we have where → stands for "continuous embedding", and C σ for the Hölder space C σ := u ∈ S (T N ) u σ := max |û 0 |, max q∈Z 2 qσ ∆ q u L ∞ < +∞ .
(2) Scaling properties: (a) For all λ > 0 and u ∈ X s (here X s stands for B s or H s ), we have u(λ·) X s (T N aλ −1 ) ≈ λ s− N   10]). Let s ∈ R, u ∈ H s resp. u ∈ B s and ψ ∈ L ∞ (R N ) be supported in the annulus C(0, R 1 , R 2 ). There exists a sequence (c q ) q∈Z such that q c 2 q ≤ 1 resp. q c q ≤ 1 and for all q ∈ Z, ψ(2 −qD u) L 2 c q 2 −qs u H s . resp. ψ(2 −qD u) L 2 c q 2 −qs u B s .
We recall the definition of the paraproduct introduced by Bony [1]. The paraproduct between u and v is defined by We thus have the following formal decomposition: uv := T u v + T v u + R(u, v), with R(u, v) := q∈Z ∆ q u∆ q v∆ q = ∆ q−1 + ∆ q + ∆ q+1 . We will use the notation T u v := T u v + R(u, v).
Let us state some estimates for the paraproduct in H s andL r T (H s ) spaces. Proposition A.4 ([10]). Let 1 ≤ r, r 1 , r 2 ≤ +∞ such that 1 r = 1 r1 + 1 r2 , then for all t ∈ R and s < N 2 , we have . If s = N 2 , the above inequalities hold true provided u H s resp. u L r 1 T (H s ) has been replaced with L ∞ resp. L r1 (0, T ; L ∞ ) . If (s, t) ∈ R 2 satisfies s + t > 0, then . Finally, for the operator L(τ ), we have the following result: