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

1.   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.

where $u$ denotes the velocity field of the fluid, $P$ the total pressure, $B$ the magnetic field, $\nu > 0$ and $\eta > 0$ are the viscosity and the magnetic diffusivity. We define the 3D periodic space domain $\Omega = [0, L]^{2}\times \mathbb{R}$, the periodic solution means $u(x+e_{i}, t) = u(x, t)\, \, (i = 1, 2, 3)$, for all $x$ and $t\geq 0$, 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 $B^{(0)} = (1, 0, 0)$. The perturbation $(u, b)$ with $b = B-B^{(0)}$ obeys,

where, for notational convenience, we write

In addition, for convenience, we define the norm for the $L^{p}(\Omega)$ space, for $p\in [1, \infty]$, is denoted by $\|f\|_{p}$. The inner product of $f$ and $g$ in the $L^{p}(\Omega)$ space is denoted by $(f, g) = \iiint_{\Omega}fg\, dx_{1}dx_{2}dx_{3}: = \int_{\Omega}fg\, dx$. Respectively, the horizontal flow is defined in $\Omega$ with $\int_{[0, L]^{2}} u\, dx = 0$ and $\int_{[0, L]^{2}} b\, dx = 0$.

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.

Theorem 1.1. Consider (1.2) with initial data $(u_0, b_0)\in H^2(\Omega)$ satisfies $\nabla\cdot u_{0} = \nabla\cdot b_{0} = 0, \int_{[0, L]^{2}} u_{0}dx = 0$ and $\int_{[0, L]^{2}} b_{0}dx = 0$. Then there exists a constant $\delta = \delta(\nu, \eta) > 0$ such that, if

then (1.2) has a unique global solution

satisfying

for any $t > 0$ and $C = C(\nu, \eta)$ 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 $H^2$-energy E(t) by

and prove that, for a constant $C > 0$ and any $t\geq 0$,

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 $\|\nabla_{h}u\|_{H^3}$ and $\|\nabla_{h}b\|_{H^3}$. 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.

2.   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.

2.1.   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 $\delta > 0$ 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 $H^{1}(\Omega) = \{f\in L^{2}(\Omega):\nabla f\in L^{2}(\Omega)\}$. In addition, we define the following Hilbert space,

that features the inner product $(f, g)_{H_{h}^{1}(\Omega)} = (f, g)_{L^{2}(\Omega)}+(\nabla_{h} f, \nabla_{h} g)_{L^{2}(\Omega)}$.

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.

Lemma 2.1. Let $f\in H^{1}(\Omega)$, $g\in H^{1}_h(\Omega)$, $h\in L^{2}(\Omega)$. Then,

The proof of Lemma 2.1 can be found in ^[9].

3.   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.

3.1.   The $H^{2}$-stability

Due to the equivalence of $\|(u, b)\|_{H^2}$ with $\|(u, b)\|_{L^2}+\|(u, b)\|_{{\dot{H}}^2}$, it suffices to bound the $L^{2}$-norm and the ${\dot{H}}^{2}$-norm of $(u, b)$. By a simple energy estimate and $\nabla\cdot u = \nabla\cdot b = 0$, we find that the $L^{2}$-norm of $(u, b)$ obeys

The rest of the proof focuses on the ${{\dot{H}}^{2}}$-norm, applying $\partial^{2}_{i}(i = 1, 2, 3)$ to (1.2) and then dotting by $(\partial^{2}_{i} u, \partial^{2}_{i} b)$, we find

where

Note that

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 $f = \partial_{i}^{k}u$, $g = \partial_{i}^{2-k}\nabla u$, $h = \partial_{i}^{2}u$ 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 $\nabla\cdot u = 0$, 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,

Also

Combining(3.6)–(3.8) yields

For $I_{4}$,

By Lemma 2.1,

We decompose $I_{42}$ into two terms

Using Lemma 2.1,

Similarly

Combining all the estimates(3.9) through (3.11) yields

It remains to estimate $I_{5}$,

By Lemma 2.1,

The difficult term is $I_{52}$, which is further decomposed into two terms

By Lemma 2.1,

Similarly, $I_{522}$ can be estimated as follows

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 $\delta > 0$, such that, if $E(0) < \delta^{2}$, then

for a pure constant C and for all $t > 0$, which implies $H^{2}$-stability.

3.2.   Uniqueness

This subsection proves the uniqueness part of Theorem 1.1. We show that two solutions $(u^{(1)}, P^{(1)}, b^{(1)})$ and $(u^{(2)}, P^{(2)}, b^{(2)})$ of (1.2) in the regularity class (1.4) must coincide. Their difference $(\widetilde{u}, \widetilde{P}, \widetilde{b})$ with

satisfies, according to (1.2)

Basic energy estimates show that

where

By Lemma 2.1, $K_{1}$, $K_{2}$, $K_{3}$, $K_{4}$ can be bounded as follows

Combining (3.16)–(3.19), we set $Y(t) = (\|\widetilde{u}(t)\|_{2}^{2}+\|\widetilde{b}(t)\|_{2}^{2})$,

where

Since $(u^{(2)}, b^{(2)})$ is in the regularity class (1.4). For any $T > 0$, we have

Gronwall's inequality applied to (3.20) implies that, for any $T > 0$,

In particular, the initial values of the two solutions in the regularity class (1.4), then (3.21) implies $Y(t) = \|\widetilde{u}(t)\|_{2}^{2}+\|\widetilde{b}(t)\|_{2}^{2}\equiv0$ for any $T > 0$. This completes the proof of the uniqueness.

4.   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.

Acknowledgments

We would like to thank the referees for giving us many helpful suggestions in improving our paper. Ji is supported by the National Natural Science Foundation of China (NSFC) under grant number 12001065.

Conflict of interest

The authors declare no conflict of interest.