Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium

Abstract

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation.

Introduction

The magnetohydrodynamic (MHD) equations reflect the basic physics laws governing the motion of electrically conducting fluids such as plasmas, liquid metals, and electrolytes. The velocity field obeys the Navier-Stokes equations with Lorentz forcing generated by the magnetic field while the magnetic field satisfies the Maxwell's equations of electromagnetism. The MHD equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering (see, e.g., [3], [14], [36]).

The MHD equations are also mathematically significant. The MHD equations share similarities with the Navier-Stokes equations, but they contain much richer structures than the Navier-Stokes equations. They are not merely a combination of two parallel Navier-Stokes type equations but an interactive and integrated system. Their distinctive features make analytic studies a great challenge but offer new opportunities.

Two fundamental problems on the MHD equations have recently attracted considerable interests. The first is the existence and uniqueness of solutions while the second concerns the stability of perturbations near physically relevant equilibrium. There have been substantial developments on these problems, especially on those MHD systems with only partial or fractional dissipation.

This paper focuses on a stability problem concerning the following 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion,

$$\{\begin{array}{c}{\partial }_{t}u+u\cdot \mathrm{\nabla }u-{\partial }_1^{2}u-{\partial }_2^{2}u+\mathrm{\nabla }P=B\cdot \mathrm{\nabla }B,\hfill \\ {\partial }_{t}B+u\cdot \mathrm{\nabla }B-{\partial }_3^{2}B=B\cdot \mathrm{\nabla }u,\hfill \\ \mathrm{\nabla }\cdot u=\mathrm{\nabla }\cdot B=0,\hfill \end{array}$$

where u represents the velocity field, P the total pressure and B the magnetic field. We provide some relevant physical backgrounds for the MHD system in (1.1). The Navier-Stokes equations with anisotropic viscous dissipation arise in several physical circumstances. It can model the turbulent diffusion of rotating fluids in Ekman layers. A standard reference is Chapter 4 of Pedlosky's book [34]. In addition, anisotropic viscous dissipation also arises in the modeling of reconnecting plasmas (see, e.g., [12], [13]). When the resistivity of electrically conducting fluids such as certain plasmas and liquid metal is anisotropic and only in the vertical direction, the vertical magnetic diffusion may be relevant (see, e.g., [35]).

It is clear that a special solution of (1.1) is given by the zero velocity field and the background magnetic field $B^{(0)}=e_1$, where $e_1=(1,0,0)$. The perturbation $(u,b)$ around this equilibrium with $b=B-e_1$ obeys

$$\{\begin{array}{c}{\partial }_{t}u+u\cdot \mathrm{\nabla }u-{\mathrm{\Delta }}_{h}u+\mathrm{\nabla }P=b\cdot \mathrm{\nabla }b+{\partial }_{1}b,\hfill \\ {\partial }_{t}b+u\cdot \mathrm{\nabla }b-{\partial }_3^{2}b=b\cdot \mathrm{\nabla }u+{\partial }_{1}u,\hfill \\ \mathrm{\nabla }\cdot u=\mathrm{\nabla }\cdot b=0,\hfill \end{array}$$

where, for notational convenience, we have written

$${\mathrm{\Delta }}_h={\partial }_1^2+{\partial }_2^2$$

and we shall also write ${\mathrm{\nabla }}_h=({\partial }_1,{\partial }_2)$.

This paper aims at the stability problem on the perturbation $(u,b)$. Equivalently, we establish a small data global well-posedness result for (1.2) supplemented with the initial condition

$$u(x,0)=u_0(x),b(x,0)=b_0(x).$$

Our main result can be stated as follows. The notation $A\lesssim D$ means $A\le CD$ for a pure constant C.

Theorem 1.1

Consider (1.2) with the initial data $(u_0,b_0)\in H^3({\mathbb{R}}^3)$ and $\mathrm{\nabla }\cdot u_0=\mathrm{\nabla }\cdot b_0=0$. Then there exists a constant $ϵ>0$ such that, if

$${\Vert u_0\Vert }_{H^3}+{\Vert b_0\Vert }_{H^3}\le ϵ,$$

then (1.2) has a unique global classical solution $(u,b)$ satisfying, for any $t>0$,

$${\Vert u(t)\Vert }_{H^3}+{\Vert b(t)\Vert }_{H^3}+\underset{0}{\overset{t}{\int }}({\Vert {\mathrm{\nabla }}_{h}u\Vert }_{H^3}^2+{\Vert {\partial }_{3}b\Vert }_{H^3}^2+{\Vert {\partial }_{1}b\Vert }_{H^2}^2)d\tau \lesssim ϵ.$$

This new result constitutes an important contribution to the stability problem on the MHD equations. Prior to this stability result, we only know the stability of the background magnetic field for two cases, the ideal MHD equations and the MHD equations with kinematic dissipation and no magnetic diffusion. The nonlinear stability for the ideal MHD equations was established in several beautiful papers [2], [6], [22], [33], [41]. The stability problem for the MHD equations with no magnetic diffusion was first studied in [31], which inspired many further investigations. The stability has now been successfully established by several authors via different approaches (see, e.g., [1], [15], [23], [24], [25], [31], [33], [37], [38], [39], [43], [44], [51]). To give a more complete view of current studies on the stability and the global regularity problems, we also mention some of the other exciting results in [9], [16], [17], [19], [20], [21], [26], [28], [29], [42], [45], [46], [47], [48], [49], [50], [52] and the references therein.

A special consequence of Theorem 1.1 and its proof is the stability or small data global well-posedness of the 3D Navier-Stokes equations with only horizontal dissipation. It is not clear if the stability for the 3D Navier-Stokes still holds if there is only one directional dissipation (say, in $x_1$ or $x_2$ direction, but not both). The 3D Navier-Stokes equations with full dissipation have small data global well-posedness while the 3D incompressible Euler equations are ill-posed and have norm inflation in any Sobolev space $H^k$ or $C^k$ for any positive integer k [4], [5], [18].

The proof of Theorem 1.1 is not trivial. A natural starting point is to bound ${\Vert u(t)\Vert }_{H^3}+{\Vert b(t)\Vert }_{H^3}$ via the energy estimates. However, due to the lack of the vertical dissipation and the horizontal magnetic diffusion, some of the nonlinear terms can not be controlled in terms of ${\Vert u(t)\Vert }_{H^3}+{\Vert b(t)\Vert }_{H^3}$ or the dissipative parts ${\Vert {\mathrm{\nabla }}_{h}u\Vert }_{H^3}$ and ${\Vert {\partial }_{3}b\Vert }_{H^3}$. Consequently we are not able to obtain a closed differential inequality for

$$E_0(t)=\underset{0\le \tau \le t}{\sup }\{{\Vert u(\tau )\Vert }_{H^3}^2+{\Vert b(\tau )\Vert }_{H^3}^2\}+2\underset{0}{\overset{t}{\int }}{\Vert {\mathrm{\nabla }}_{h}u(\tau )\Vert }_{H^3}^2+{\Vert {\partial }_{3}b(\tau )\Vert }_{H^3}^{2}d\tau .$$

This forces us to include suitable extra terms in the energy estimates. We discover that the following term

$$E_1(t)=\underset{0}{\overset{t}{\int }}{\Vert {\partial }_{1}b(\tau )\Vert }_{H^2}^{2}d\tau$$

serves our purpose perfectly. All nonlinear terms involved in the estimates of $E_0(t)$ can be bounded in terms of $E_0(t)$ and $E_1(t)$. The selection of this term is based on the structure of (1.2) and through trial and error. We remark that the process of estimating $E_0(t)$ involves many terms and is very lengthy. Even with the combination of $E_0(t)$ and $E_1(t)$, it is still very difficulty to directly bound some of the nonlinear terms. Two of the most difficult ones are

$$\underset{{\mathbb{R}}^3}{\int }{\partial }_1{u}_1{\partial }_2^3{b}_1{\partial }_2^3{b}_{3}dx\text{and}\underset{{\mathbb{R}}^3}{\int }{\partial }_1{u}_3{\partial }_2^3{b}_1{\partial }_2^3{b}_{3}dx.$$

It does not appear possible to bound them directly in terms of $E_0(t)$ and $E_1(t)$. Our strategy is to make use of the special structure of the equation for b in (1.2) and replace ${\partial }_1{u}_1$ and ${\partial }_1{u}_3$ in (1.3) via the equation of b,

$${\partial }_{1}u={\partial }_{t}b+u\cdot \mathrm{\nabla }b-{\partial }_3^{2}b-b\cdot \mathrm{\nabla }u.$$

Substituting (1.4) in (1.3) generates more terms, but fortunately all the resulting terms can be bounded suitably by $E_0(t)$ and $E_1(t)$.

In addition, in order to make most efficient usage of the anisotropic dissipation, we employ extensively the following anisotropic bounds in the estimates of the nonlinear terms. These anisotropic bounds are extremely powerful in the study of global regularity and stability problems on partial differential equations with only partial dissipation. Similar inequalities have previously been used in the investigation of partially dissipated 2D MHD systems and related equations (see, e.g., [7], [8]).

Lemma 1.2

The following estimates hold when the right-hand sides are all bounded.

$$\begin{gathered} \underset{{\mathbb{R}}^3}{\int }|fgh|dx\lesssim {\Vert f\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{1}f\Vert }_{L^2}^{\frac{1}{2}}{\Vert g\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{2}g\Vert }_{L^2}^{\frac{1}{2}}{\Vert h\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{3}h\Vert }_{L^2}^{\frac{1}{2}},\underset{{\mathbb{R}}^3}{\int }|fghv|dx\lesssim {\Vert f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{1}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{2}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_1{\partial }_{2}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert g\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{1}g\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{2}g\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_1{\partial }_{2}g\Vert }_{L^2}^{\frac{1}{4}} \\ \cdot {\Vert h\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{3}h\Vert }_{L^2}^{\frac{1}{2}}{\Vert v\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{3}v\Vert }_{L^2}^{\frac{1}{2}},{(\underset{{\mathbb{R}}^3}{\int }|fgh{|}^{2}dx)}^{\frac{1}{2}}\lesssim {\Vert f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{1}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{2}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_1{\partial }_{2}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert g\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{3}g\Vert }_{L^2}^{\frac{1}{2}}{\Vert h\Vert }_{H^2},\underset{{\mathbb{R}}^3}{\int }|fgh|dx\lesssim {\Vert f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{1}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_{2}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert {\partial }_1{\partial }_{2}f\Vert }_{L^2}^{\frac{1}{4}}{\Vert g\Vert }_{L^2}^{\frac{1}{2}}{\Vert {\partial }_{3}g\Vert }_{L^2}^{\frac{1}{2}}{\Vert h\Vert }_{L^2}. \end{gathered}$$

Combining all aforementioned ingredients, we are able to drive the following energy inequalities

$$E_0(t)\lesssim E_0(0)+E_0{(0)}^{\frac{3}{2}}+E_0{(t)}^{\frac{3}{2}}+E_1{(t)}^{\frac{3}{2}}+E_0{(t)}^2+E_1{(t)}^2$$

and

$$E_1(t)\lesssim E_0(0)+E_0(t)+E_0{(t)}^{\frac{3}{2}}+E_1{(t)}^{\frac{3}{2}}.$$

These inequalities, combined with the bootstrapping argument, allow us to prove Theorem 1.1.

We remark that many important results on the stability problem concerning the 3D anisotropic Navier-Stokes equations with horizontal dissipation have been obtained (see, e.g., [10], [11], [27], [32]). Since the equation of the magnetic field involves only vertical dissipation, the situation here is different from two parallel Navier-Stokes equations with horizontal dissipation. In fact, the stability problem on the 3D Navier-Stokes equation with only vertical dissipation remains an outstanding open problem. The stability on the MHD system studied here is only made possible by fully exploiting the stabilizing effects of the background magnetic field and making use of the hidden structure in the MHD system.

The rest of this paper is divided into three sections. Section 2 provides the proofs of Theorem 1.1 and of Lemma 1.2. Section 3 derives the energy inequality (1.5) while Section 4 proves (1.6).