Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class

We study the zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic (MHD) equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the resistivity coefficient, for the solutions to the viscous incompressible MHD equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible MHD equations converge to that of the ideal incompressible MHD equations as the viscosity and resistivity coefficients go to zero. Moreover, the convergence rate is also given.

The system (1) describes the macroscopic behavior of electrically conducting incompressible fluids in a given magnetic field. Due to the significance of the physical background, the incompressible MHD equations have been studied by many physicists and mathematicians on various topics, for example, see [6,11,18,15] and the references cited therein. One of the most important problems in magnetohydrodynamics is to understand the inviscid limit in a domain with boundary. The outcomes of the inviscid limit and convergence rate are helpful to understand the turbulent When the viscous incompressible MHD equations are supplemented with Navier boundary conditions, the authors in [28,47] studied the inviscid limit problem in anisotropic conormal Sobolev spaces and classical Sobolev spaces, respectively. The inviscid limit problem of the viscous incompressible MHD equations with non-slip boundary condition for velocity is very challenging due to the appearance of nontrivial boundary layer. Recently, Liu, Xie, and Yang [31] studied the well-posedness theory of the MHD boundary layer equations in weighted Sobolev spaces in 2D case. In addition, based on the well-posedness theory of the MHD boundary layer equations, they [32] also studied the inviscid limit. Wang and Xin [45] investigated the zero viscosity-resistivity limit to the incompressible MHD equations by constructing the boundary layer directly.
In this paper, motivated by [7] on the incompressible Navier-Stokes equations, we study the zero viscosity-resistivity limit of the viscous incompressible MHD equations in the torus T 3 in Gevrey class. Due to the strong coupling between u ν,µ and H ν,µ , we need to estimate some new nonlinear terms. In order to overcome the difficulty in handling the nonlinear terms, we shall establish a more general result (see Lemma 2.2 below) than that in Lemma 2.3 of [7]. Besides, we also need to construct more complicated energy estimates to obtain the desired convergence results. The details will be presented in the proceeding arguments.
Before giving a rigorous justification for the above formal procedure and stating our main results, we first introduce the functions spaces used throughout this paper. Let L 2 (T 3 ) be the vector function space whereû k is the k-th order Fourier coefficient of u and i = √ −1. The condition k ·û k = 0 implies that ∇ · u = 0 in the weak sense, so it is the standard L 2 space with the divergence-free condition. Let H r (T 3 ) be the vector periodic Sobolev space: for r ≥ 1, where the condition k ·û k = 0 also means ∇ · u = 0, so it is the standard Sobolev space H r with the divergence-free condition. Denote (·, ·) the L 2 inner product of two vector functions. Let us define the fractional differential operator Λ = (−∆) 1 2 and the exponential operator e τ Λ 1 s as follows,

FUCAI LI AND ZHIPENG ZHANG
The vector Gevrey space G s r,τ (s ≥ 1, We point out that the Gevrey space is an intermediate space between the spaces of smooth functions and analytic functions. The name is given in honor of M. Gevrey, who gave the first motivating example, refer to [19], in which the regularity estimates of the heat kernel were deduced. Thanks to [39], we know that when s = 1, this space is the same as the space of analytic functions, and for s > 1, there are non-trivial Gevrey functions having compact support, which is different from analytic functions. Foias and Temam [13] first used the Fourier space method, namely the Gevrey class regularity, to prove the analyticity of the solutions for the incompressible Navier-Stokes equations. Later, the techniques developed in [13] were improved and used to other problems, see [2,26,27] and the references cited therein. Now our main results can be stated as follows. Then there exist a time T > 0, a constant C 0 > 0, and a decreasing function τ (t) > 0, independent of µ, ν ∈ (0, 1] such that, for any fixed ν, µ, there exists a unique solution (u ν,µ , H ν,µ , p ν,µ ) of the problem (1) As for the problem (3)-(4), there also exists a unique solution (u 0 , H 0 , p 0 ) on [0, T ] satisfying Moreover, we have the following convergence rate: for any 0 < t ≤ T , where C 1 is a constant depending on r, s, u 0 , H 0 and T .
Remark 1.1. The uniform Gevrey radius τ (t) of the solution is where C 2 and C 3 are two positive constants depending on r, s, u 0 , H 0 and T .
The rest of this paper is organized as follows. In Section 2, we list a local existence result of the H r solution to the problem (1)-(2), and some elementary inequalities which will be used later. Next, we derive a priori estimates in Gevrey space, and use the a priori estimates to prove the existence of the solution in Gevrey space in Section 3. In Section 4, we show the convergence rate of the zero viscosity-resistivity limit of the problem (1)-(2) in Gevrey space. Finally, in the Appendix, we give the proof of the existence of the H r solution.

Preliminaries.
We first give a result on the local existence of the H r solution to the viscous incompressible MHD equations.
The proof of Theorem 2.1 will be given in the Appendix. As for the ideal MHD equations, by taking the similar arguments to that in Theorem 2.1, we can also obtain the local existence of the H r solution in T 3 . Next, we give two important inequalities, which will be used repeatedly throughout this paper. The first one is that, for any j, k ∈ Z 3 \{0}, The proof is a simple application of triangle inequality. So we omit the details here. The second one is the following lemma.
). Given two real numbers ξ, η ≥ 1 and s ≥ 1, then the following inequality holds where C > 0 is a constant depending only on s.
Finally, with the help of (13) and Lemma 2.1, we have the following estimate which will be used to estimate the nonlinear terms.
Lemma 2.2. Let r > 0, s ≥ 1 and τ > 0 be given constants and r > max 5 2 + 2 s , 7 2 + 1 2s . Then, for any f, g, h ∈ G s where the constant C depends only on r and s.
Proof. It follows from the definition of the vector function space G s andf 0 =ĝ 0 =ĥ 0 = 0. In view of the Fourier series convolution property, we have

FUCAI LI AND ZHIPENG ZHANG
Applying Λ r e τ Λ 1 s to f · ∇g, we obtain that In addition, Λ r e τ Λ 1 s h = l∈Z 3 |l| r e τ |l| 1 sĥ l e il·x . Now we take the L 2 inner product of Λ r e τ Λ 1 s (f · ∇g) with Λ r e τ Λ 1 s h over T 3 . The orthogonality of the exponentials in L 2 yields Similarly, we also have Then, (18) minus (19) yields where Before beginning to control I 1 and I 2 , we recall the following mean value theorem, for any ξ, η ∈ R + , there exist two constants 0 ≤ θ, θ ≤ 1 such that which implies that where the constant C depends only on r. On the other hand, the inequality e ξ ≤ e + ξ 2 e ξ holds for all ξ ∈ R, which yields Then it follows from (22) and (23) that where In view of the discrete Hölder's inequality and Minkowski's inequality, we can obtain the estimates of I 1i (i = 1, . . . , 5). For the first term I 11 , we have where C is a constant depending on r and e. Arguing analogously to I 11 , we deduce that

FUCAI LI AND ZHIPENG ZHANG
where C is a constant depending on r, s and e, and the assumption r > 5 2 + 2 s has been used to estimate I 13 . As for I 14 , using the facts s ≥ 1 and r > 7 2 + 1 2s , and the inequality |k − j| 1 2s ≤ C|k| 1 2s |j| 1 2s , we obtain that Hence, substituting (25)-(29) into (24), we obtain that For the term I 2 , we first have the following observations: Due to s ≥ 1, we have which, together with (31), imply that Additional, in view of Lemma 2.1, we have Therefore, it follows from (33) and (34) that where Here we have used the inequalities and e ξ ≤ 1+ξe ξ for ξ ∈ R + . Using the discrete Hölder's inequality and Minkowski's inequality, we conclude that Thus, we derive from (35)-(37) that Finally, inserting (30) and (38) into (20), we obtain (15). This completes the proof of Lemma 2.2.
3. Uniform regularity of the solutions. In this section, we prove the existence of Gevrey class solutions (u ν,µ , H ν,µ ) to the viscous incompressible MHD equations. And the existence of Gevrey class solution (u 0 , H 0 ) to the ideal MHD equations can be verified similarly. The methods of our proof are based on Galerkin approximation. To this end, we introduce the following equivalent problem for (1)- (2): where A = −P∆ is the well-known Stokes operator and P is Leray projector. Similarly, we have the following equivalent problem for (3)-(4): We first recall some properties of the Stokes operator A.  1 ([9]). The Stokes operator A is symmetric and self-adjoint, moreover, the inverse of the Stokes operator, A −1 , is a compact operator in L 2 . The Hilbert theorem implies that there exists a sequence of positive numbers λ j and an orthonormal basis ω j of L 2 , which satisfies Aω j = λ j ω j , 0 < λ 1 < .... < λ j ≤ λ j+1 ≤ ..., lim j→∞ λ j = ∞.

FUCAI LI AND ZHIPENG ZHANG
Particularly, in the case of T 3 , the eigenvector functions and the eigenvalues have the following definite expression where k = (k 1 , k 2 , k 3 ) ∈ Z 3 , k = 0, j = 1, 2, 3 and {e j } j=1,2,3 are the canonical basis in R 3 . Hence, we find that each ω j is not only in L 2 , but also in G s r,τ for any r > 0. Now we will show that there exists a solution to (39) with the initial data (u 0 , H 0 ) ∈ G s r,τ , where s ≥ 1, r > max 5 2 + 2 s , 7 2 + 1 2s , and τ (t) > 0 is a differentiable decreasing function of t. For this purpose, we first derive the following a priori estimate which is the crucial step in the proof of Theorem 1.1. For notational convenience, we shall drop the superscripts ν and µ throughout this section.
) be the solution to (39). Then, for any 0 < t ≤ T , Under the same assumptions as above, let (u 0 , H 0 ) ∈ L ∞ (0, T ; G s r,τ (·) (T 3 )) be the solution to (40), then we have Moreover, the uniform radius τ (t) is given by where C 2 , C 3 , C 1 T and C 2 T are positive constants depending on u 0 , H 0 , r, s and T .
Proof. Applying Λ r e τ Λ 1 s to both sides of (39) 1 and taking the L 2 inner product of both sides with Λ r e τ Λ 1 s u, we have where we have used the facts that P commutes with Λ r e τ Λ 1 s and P is symmetric. Arguing analogously to (46), we can also obtain that 1 2 Adding (46) and (47) up gives 1 2 where s H , s u , Now we come to estimate the right-hand side terms of (48). First, in view of the incompressible condition of H, we infer that

It follows from Lemma 2.2 that
Next, using Lemma 2.2 straightforwardly, we have Finally, due to the incompressible condition of u, we find that By virtue of Lemma 2.2, we have Now if the radius of Gevrey class τ (t) is smooth and decreasing fast enough such that the following inequality holds, Then we obtain that 1 2 Furthermore, since µ, ν > 0 and by Cauchy-Schwarz inequality, we infer that An application of Gronwall's inequality to (56) then implies that, for 0 < t < T , Moreover, it follows from (12) that the H r solution satisfies the following inequality: for 0 < t < T , which, together with (57), yields (59) Note that a sufficient condition for (54) to hold is that By solving the ordinary differential equation (60), we get . (61) Resetting the constants in (61), we obtain that where C 2 and C 3 are constants depending on r, s, T, u 0 and H 0 . Then we have proved (11) in Remark 1.1. Integrating (55) in time from 0 to t, we have, for where C 2 T depends on r, s, T, u 0 , H 0 and τ 0 . It should be noted that all of the above estimates are independent of ν and µ. Therefore, by letting ν = µ = 0 in (53) and proceeding exactly as above, we can obtain similarly a priori estimates for the solution (u 0 , H 0 ) to (40). Hence the proof of Proposition 3.2 is completed.
Next, with the help of the a priori estimates obtained in Proposition 3.2, we use the Galerkin approximation, following the arguments in [9], to establish the local existence of the solutions (u, H) and (u 0 , H 0 ) in Gevrey class G s r,τ . Proof. We start with a sequence of approximate functions (u (m) , H (m) ): where {ω j } ∞ j=1 are the orthonormal basis given in Proposition 3.1, and u j,m and H j,m (j = 1, 2, ..., m) solve the following differential equations: with b(ω k , ω l , ω j ) = T 3 ω k · ∇ω l · ω j dx. In view of the standard nonlinear ordinary differential equation theory, the problem (65) is locally well-posed, say on [0, T m ). Now, we need to show that T m can be extended to T . Taking the inner products ((65) 1 , u j,m ) and ((65) 2 , H j,m ) (j = 1, 2, ..., m), adding them up, and using the facts that which yields that, for 0 < t ≤ T m , Therefore, for every T m , it can be extended to T , and (u (m) , H (m) ) is bounded in L ∞ (0, T ; L 2 ), uniformly for m.
Moreover, for any t ∈ [0, T ), (u (m) , H (m) ) solves the following system where χ m denotes the orthogonal projector from L 2 into the space spanned by ω j m j=1 . Then, we turn to obtain the uniform Gevrey class norm bound for (u (m) , H (m) ). To this end, we first apply Λ r e τ Λ 1 s to (68) 1 and (68) 2 , take the inner product with Λ r e τ Λ 1 s u (m) and Λ r e τ Λ 1 s H (m) , respectively, and then use the properties that the operators χ m and P commute with Λ r e τ Λ 1 s to obtain that Arguing analogously to Proposition 3.2, we have Thus (u (m) , H (m) ) is bounded in L ∞ (0, T ; G s r,τ ), uniformly for m, In order to pass to the limit in the nonlinear terms by using compactness arguments, we need to give the temporal derivative estimate of (u (m) , H (m) ). From (68), we have Then, it follows from Theorem 2.1 and (73) that Based on Aubin-Lions-Simon theorem in [14] and the fact that H r is compactly embedded in L 2 , we prove that there exists a subsequence (u (m l ) , Choosing v ∈ L 2 arbitrarily, taking the inner product of (68) 1 and (68) 2 with v and integrating in time, we obtain that

FUCAI LI AND ZHIPENG ZHANG
According to (75) and (76) and passing to the limit in (77), we have So (u, H) is a weak solution to (39). Furthermore, due to the lower semicontinuity of norms, (u, H) is the H r solution to (39). Now, we pay attention to the regularity of (u, H) in Gevrey class. The arguments are similar to that in [27]. We first give a uniform bound for ( d dt u (m l ) , d dt H (m l ) ) in Gevrey class. To this end, we recall an inequality that f g G s r,τ ≤ C r f G s r,τ g G s r,τ for s ≥ 1, whose proof is similar to Lemma 1 of [13]. It follows from (68) that Thus, by virtue of (71) and (72), we can verify that d dt u (m l ) is uniformly bounded in L 2 (0, T ; G s r−1,τ ) with respect to l. Arguing analogously to (79), we also have that d dt H (m l ) is uniformly bounded in L 2 (0, T ; G s r−1,τ ) with respect to l. Let 0 < 1. Thanks to the compact embedding G s r,τ → G s r− ,τ and the above bounds, we can prove by the Aubin-Lions-Simon Theorem that there exists a subsequence of (u (m l k ) , H (m l k ) ) ∞ k=1 , which converges to an element (v, B) in C(0, T ; G s r− ,τ ). As shown in (76), this sequence also converges to (u, H) in C(0, T ; L 2 ). By the uniqueness of limits, (u, H) = (v, B) ∈ C(0, T ; G s r− ,τ ). Furthermore, we also have Since this bound holds uniformly for all 0 < 1, it also holds for = 0, which can be proved by using the Fourier series representation taking the limit as → 0, and passing the limit inside the infinite sum by the Levi Monotone Convergence theorem for the counting measure. The uniqueness of the solution is clear since we work with functions with Lipschitz regularity. Consequently, we obtain the first conclusion in Theorem 3.1.
For the ideal MHD equations, according to the similar approaches described above, we can obtain the existence of solution in Gevrey class space and we omit the details here. Thus we complete the proof of Theorem 3.1.
It remains to show that (u, H) is the solution of (1) and (2). In fact, based on Theorem 3.1, we have Thus there exists a scalar function p such that where p is unique up to a constant, and p satifies with periodic boundary condition. For the regularity of the pressure p in Gevrey class space, we have the following proposition.
Proposition 3.3. Let p satisfies (82), then the following estimate holds, where T and C 1 T are defined in Proposition 3.2. And for the pressure p 0 in (3), we also have Proof. To begin with, from the standard elliptic equations theory, the existence of the pressure p in H r+1 can be guaranteed. Now we focus on the regularity of p The first term of the right-hand side of (85) can be bounded by where the constant C is independent on m. For the second term of the right-hand side of (85), taking the same arguments as those in (86), we have Then, substituting (86) and (87) into (85), we obtain that We first note that when m = 1, p G s r+1,0 ≤ p H r+1 . So, by the induction on m, (88) and the fact p ∈ H r+1 , we can verify that p ∈ G s r+1, 2 m −1 2 m for any positive integer m.
Now we turn to prove (83). Arguing analogously to (86), we can also obtain that where C is a constant independent on m. It follows from (59) that Since the bound in (90) holds uniformly for all positive integer m, using the definition of · G s r+1, 2 m −1 2 m τ in (6) and the Levi Monotone Convergence theorem for the counting measure, we have that Therefore, (83) holds. For the pressure p 0 (t, x) in the ideal MHD equations, we can first obtain the following elliptic equation Then by using the same arguments as above, we can get (84).
4. Zero viscosity-resistivity limit. In this section, we will show the zero viscosity-resistivity limit of the viscous incompressible MHD equations in Gevrey class space. Moreover, we give the convergence rate with respect to µ and ν.
Theorem 4.1. Let (u ν,µ , H ν,µ , p ν,µ ) and (u 0 , H 0 , p 0 ) are the solutions obtained in the previous section. Then the following inequalities hold, for any 0 < t ≤ T , where C 1 is a constant depending on r, s, u 0 , H 0 and T .
In addition, s ≥ 1 implies that e τ |k|

FUCAI LI AND ZHIPENG ZHANG
By the discrete Hölder's inequality and Minkowski's inequality, we obtain that For T 5 , T 7 and T 9 , by taking the same arguments as those to T 3 , we get As for T 4 , we can rewrite it as where Then it follows from (108) that For T 2 4 , by the inequality |e ξ − 1| ≤ |ξ|e |ξ| for ξ ∈ R and Lemma 2.1, we obtain that