Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain

Abstract: The stability problem on the magnetohydrodynamics (MHD) equations with partial or no dissipation is not well-understood. This paper focuses on the 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the stability of perturbations near the steady solution given by a background magnetic field in periodic domain. The new stability result presented here is among few stability conclusions currently available for ideal or partially dissipated MHD equations.


Introduction
In recent few years, there have been substantial developments concerning the MHD equations, especially there is only partial or fractional dissipation. The MHD equations govern the motion of electrically conducting fluids such as plasmas, liquid metals, and electrolytes. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. The set of equations that describe MHD are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. Since their initial derivation by the Nobel Laureate H. Alfvén [1] in 1924, the MHD equations have played vital roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering (see, e.g., [2,3]). This paper establishes the stability of perturbations near a background magnetic field of the 3D MHD equations with mixed partial dissipation and magnetic diffusion in periodic domain.
∂ t u + u · ∇u = −∇P + ν ∆ h u + B · ∇B, x ∈ Ω, t > 0, ∂ t B + u · ∇B = η∆ h B + B · ∇u, x ∈ Ω, t > 0, ∇ · u = ∇ · B = 0, x ∈ Ω, t > 0, (1.1) where u denotes the velocity field of the fluid, P the total pressure, B the magnetic field, ν > 0 and η > 0 are the viscosity and the magnetic diffusivity. We define the 3D periodic space domain Ω = [0, L] 2 × R, the periodic solution means u(x + e i , t) = u(x, t) (i = 1, 2, 3), for all x and t ≥ 0, where e i are the standard basis vectors, e 1 = (1, 0, 0) t . We know that (1.1) admits the following steady state solution It is clear that a special solution of (1.1) is given by the zero velocity field and the background magnetic fields where, for notational convenience, we write In addition, for convenience, we define the norm for the L p (Ω) space, for p ∈ [1, ∞], is denoted by f p .
The inner product of f and g in the L p (Ω) space is denoted by ( f, g) = Ω f g dx 1 dx 2 dx 3 := Ω f g dx.
Respectively, the horizontal flow is defined in Ω with [0,L] 2 u dx = 0 and [0,L] 2 b dx = 0. This paper aims at the stability problem on the perturbation of (1.1) near (u (0) , B (0) ). Equivalently, we establish a small data global well-posedness result for (1.2) supplemented with the initial condition Our main result can be stated as follows.
then (1.2) has a unique global solution for any t > 0 and C = C(ν, η) is a constant.
The MHD equation, especially those with partial dissipation have recently attracted considerable interests. There are substantial developments on two fundamental problems, the global regularity and stability problems, which have been successfully established by many authors via different approaches [4][5][6][7]. In particular, it is also worth mentioning the beautiful work of [8], which made further progress by providing the stability of perturbations near a background magnetic field of the 3D incompressible MHD equation with mixed partial dissipation and deal with the H 3 -estimate. To give a more complete views of current studies on the stability, we also mention some of exciting results in [9][10][11][12]. In this paper, we mainly deal with the H 2 -estimate for the solution of (1.2). The stability of the incompressible MHD equation with mixed partial dissipation is not well-solved, except in the periodic case. Our study of the stability problem on (1.2) is inspired by the recent important result in [13], which is different with the whole region is that helps to solve the periodic problem.
We employ the bootstrapping argument to prove the desired H 2 -stability. And we define the H 2energy E(t) by and prove that, for a constant C > 0 and any t ≥ 0, Once (1.6) is established, an application of the bootstrapping argument would imply the desired global stability. The details are given in section 2. Due to the presence of the anisotropic dissipation, we make use of anisotropic estimates for triple products (see Lemma 2.1 in section 2).
The proof of Theorem 1.1 is not trivial. A natural starting point is to bound u H 3 + b H 3 via the energy estimates. However, due to the lack of the vertical dissipation, some of the nonlinear terms can not be controlled in terms of u H 3 + b H 3 or the dissipation parts ∇ h u H 3 and ∇ h b H 3 . Thus, we show the stability of equations (1.1) by bootstrapping argument which will be shown in section 3, and we also show the uniqueness in that section.

Proof of Theorem 1.1 and anisotropic estimates
This section applies the bootstrapping argument to prove Theorem 1.1. In addition, we provide the anisotropic inequality to be used in the proof of (2.1) in the subsequent section.

Proof of Theorem 1.1
Roughly speaking, the bootstrapping argument starts with an ansatz that E(t) is bounded, say and show that E(t) actually admits a smaller bound, say when the initial condition is sufficiently small. A rigorous statement of the abstract bootstrapping principle can be found in T. Tao's book (see [14]). To apply the bootstrapping argument to (2.1), we assume that When (2.1) holds, we have It then follows from (1.6) that if we choose δ > 0 sufficiently small such that

then (1.3) and (2.2) imply that
the bootstrapping argument then leads to the desired global bound this completes the proof of Theorem 1.1. As usual, the Sobolev space In addition, we define the following Hilbert space, (Ω) . The rest of this section provides the anisotropic inequality. The MHD system examined in this paper involves the estimates of quite a few triple terms. Anisotropic inequality appears to be necessary to deal with such partially dissipated system.
The proof of Lemma 2.1 can be found in [9].

Proof of kernal part of bootstrapping
This section proves the major estimate in (1.6), namely where E(t) is defined in (1.5). The core of the proof is to bound the H 2 -norm of (u, b) suitably. For the sake of clarity, the proof is divided to two main parts, the first one is devoted to the H 2 -stability and the second one is to the uniqueness. The local existence can be obtained by a standard approach of Friedrichs' method of cutoff in Fourier space (see, e.g., [15]), we omit the details here.

The H 2 -stability
Due to the equivalence of (u, b) H 2 with (u, b) L 2 + (u, b) Ḣ2 , it suffices to bound the L 2 -norm and theḢ 2 -norm of (u, b). By a simple energy estimate and ∇ · u = ∇ · b = 0, we find that the L 2 -norm of (u, b) obeys The rest of the proof focuses on theḢ 2 -norm, applying ∂ 2 i (i = 1, 2, 3) to (1.2) and then dotting by where Integrating by parts and u(x + e i , t) = u(x, t) (i = 1, 2, 3), I 1 = 0. To bound I 2 , we decompose it into two pieces I 21 involves the favorable partial derivatives in x 1 and x 2 , respectively. Its handling is not difficult. In contrast, I 22 has partial in terms of x 3 and the control of I 22 is delicate. By Lemma 2.1 with f = ∂ k i u, g = ∂ 2−k i ∇u, h = ∂ 2 i u and Poincaré's inequality, we obtain where we have used the Poincaré's inequality We further decomposed I 22 into two terms By Lemma 2.1, Using Lemma 2.1 and ∇ · u = 0 , we obtain Combining (3.3)-(3.5), we find We now turn to the estimates of I 3 , By Lemma 2.1, Similar to I 22 , I 32 is naturally split into two terms By Lemma 2.1, Combining(3.6)-(3.8) yields By Lemma 2.1, (3.9) We decompose I 42 into two terms Using Lemma 2.1, (3.10) Similarly (3.11) Combining all the estimates(3.9) through (3.11) yields It remains to estimate I 5 , By Lemma 2.1, (3.12) The difficult term is I 52 , which is further decomposed into two terms By Lemma 2.1, (3.13) Similarly, I 522 can be estimated as follows (3.14) Combining (3.12)-(3.14), we have Therefore, if we set By Hölder's inequality, the time integral of the bounds for I 2 , I 3 , I 4 and I 5 can be estimated as follows Integrating (3.2) in time and combining with (3.1), we find A bootstrapping argument implies that, there is δ > 0, such that, if E(0) < δ 2 , then for a pure constant C and for all t > 0, which implies H 2 -stability.

Conclusions
In this paper, we gave the stability of the 3D incompressible MHD equations near a background magnetic field with horizontal dissipation in periodic domain by bootstrapping argument. The main part of bootstrapping argument relies on proof of inequality (2.1). We get through it by the anisotropic inequality, and Poincaré's inequality helps a lot in periodic domain.