Stability for the 2D MHD equations with horizontal dissipation

Abstract

In this paper we consider the following 2D MHD system with horizontal dissipation in a strip domain $T \times R$ . $\begin{matrix} \{\begin{matrix} \partial_{t} u + u \cdot \nabla u + \partial_{11} u + \nabla p = b \cdot \nabla b, \\ \partial_{t} b + u \cdot \nabla b + \partial_{11} b = b \cdot \nabla u, \\ \nabla \cdot u = \nabla \cdot b = 0, \\ u (0) = u_{0} (x), b (0) = b_{0} (x) . \end{matrix} \end{matrix}$ A bootstrapping argument together with a more accurate energy functional is employed in order to get the stability for the above system. Moreover, using a suitable transform, we also investigate the 2D MHD system with vertical dissipation in a strip domain $R \times T$ .

Keywords

2D MHD system horizontal dissipation vertical dissipation strip domain

1. Introduction and main results

In this paper, we will consider the following 2D MHD equations with horizontal dissipation $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u + u \cdot \nabla u + \partial_{11} u + \nabla p = b \cdot \nabla b, \\ \partial_{t} b + u \cdot \nabla b + \partial_{11} b = b \cdot \nabla u, \\ \nabla \cdot u = \nabla \cdot b = 0, \\ u (0) = u_{0} (x), b (0) = b_{0} (x) . \end{matrix} \end{matrix}$

We are interested in the existence of the global solution and stability of solution for the system with only horizontal dissipation. The MHD equations describe the motion of electrically conducting fluids in the presence of a magnetic field such as plasmas, electrolytes [7,15]. The MHD equations are a combination of the Navier–Stokes equations and Maxwell’s equations of electro-magnetism, that is, when we set $b = 0$ in the MHD system, it just become a standard Navier–Stokes equations.

It is worth to point out that the MHD system has richer structures than the Navier–Stokes equations, the distinctive features of which makes mathematical studies a greater challenge but offer a new opportunities at the same time. For more details about the physical backgrounds, we refer the readers to [1–4,12].

The pioneering work on the well-posedness of the fully dissipative MHD equations goes back to papers of Duvaut and Lions [11] and Sermange and Teman [25]. There are substantial recent developments about the well-posedness problem and stability of the MHD systems.

Cao and Wu [6] investigate the following 2D anisotropic MHD equations $\begin{matrix} \{\begin{matrix} \partial_{t} u + u \cdot \nabla u - ν_{1} \partial_{11} u - ν_{2} \partial_{22} u + \nabla p = b \cdot \nabla b, \\ \partial_{t} b + u \cdot \nabla b - η_{1} \partial_{11} b - η_{2} \partial_{22} b = b \cdot \nabla u . \end{matrix} \end{matrix}$ By establishing a important anisotropic inequality, they proved the global well-posedness for the case: $ν_{1} > 0$ , $η_{2} > 0$ and the case: $ν_{2} > 0$ , $η_{1} > 0$ . The 2D MHD equations with horizontal dissipation were also investigated by Cao, Regmi and Wu in [5], where they proved that the horizontal component of any solution admits a global (in time) bound in any Lebesgue space $L^{2 r}$ with $1 ⩽ r < \infty$ .

The stability of perturbation near a background magnetic filed is a interesting topic. Lin and Wu [18] studied the stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation. In recent paper [20], Nicki, Lin and Wu investigated the stability of the 2D incompressible MHD equations with magnetic filed and damping. The main ingredient to solve these problems is that the linearization of the perturbation systems have the same wave structure.

Recently, Wu and Zhu [30] obtained the stability of background solution about 3D MHD equations with mixed partial dissipation. It is natural to ask that whether the horizontal diffusion MHD system without perturbations has a global smooth solution, in particular, whether the 2D MHD with magnetic field dissipation has a global solution? In recent paper [27], Wei and Zhang proved the global well-posedness for the 2D MHD equations with magnetic diffusion by observing that the $H^{1}$ norm and $L^{\infty}$ norm of b has exponential decay in torus. Their results were improved recently by Ye and Yin in [31]. The MHD equations with only partial or fractional dissipation is also an interesting topic, we bring the reader’s attention to the papers [13,14,16,17,19,22–24,28–30] for related results.

Recently, Dong et al. [9] studied the stability and the precise large-time behavior of perturbations near the hydrostatic equilibrium of the 2D Boussinesq equations with only horizontal dissipation. The key observation is that the oscillation of u enjoys a strong version of the Poincaré type inequality in the strip domain. Particularly, the equations for the perturbation $(u, p, θ)$ satisfies $\begin{matrix} (1.2) & \{\begin{matrix} \partial_{t} u + u \cdot \nabla u + \nabla p = \partial_{11} u + θ e_{2}, \\ \partial_{t} θ + u \cdot \nabla θ + u_{2} = κ \partial_{11} θ, \\ \nabla \cdot u = 0, \\ u (x, 0) = u_{0} (x), θ (x, 0) = θ_{0} (x) . \end{matrix} \end{matrix}$ Compared with the original equations, the perturbation system (1.2) just add one term $u_{2}$ which is used to eliminate the $θ e_{2}$ in their proofs. In [8], Deng, Wu and Zhang investigated the stability of the Couette flow for 2D Boussinesq system with vertical dissipation.

Motivated by the above works, we will study the interesting problem about the well-posedness of 2D MHD equations with horizontal dissipation in this paper. Due to the lack of dissipation in the vertical direction, it is difficult to control the nonlinear term, such as $u \cdot \nabla u$ , $b \cdot \nabla b$ . Applying the Lemma 2.2 in [9], a strong Poincaré inequality and a more elaborate global energy inequality, we show the stability of the 2D MHD equations with horizontal dissipation in a strip domain by bootstrap argument.

Theorem 1.1.
Let $T = [0, 1]$ be a 1 D periodic box and $Ω = T \times R$ . Suppose that $(u_{0}, b_{0}) \in H^{2} (R^{2})$ satisfy $\nabla \cdot u_{0} = 0$ , $\nabla \cdot b_{0} = 0$ . Then there exists $δ > 0$ such that, if $\begin{matrix} ‖ u_{0} ‖_{H^{2}} + ‖ b_{0} ‖_{H^{2}} ⩽ δ, \end{matrix}$ then ( 1.1 ) has a unique global solution $(u, b) \in C ([0, \infty); H^{2} (R^{2}))$ satisfying $\begin{matrix} {‖ (u (t), b (t)) ‖}_{H^{2}}^{2} + \int_{0}^{t} {‖ \partial_{1} u (s) ‖}_{H^{2}}^{2} d s + \int_{0}^{t} {‖ \partial_{1} b (s) ‖}_{H^{2}}^{2} d s ⩽ C δ^{2}, \end{matrix}$ for any $t > 0$ and some uniform constant C.

For the following 2D MHD equations with vertical dissipation, we have a similar stability result. $\begin{matrix} (1.3) & \{\begin{matrix} \partial_{t} u + u \cdot \nabla u + \partial_{22} u + \nabla p = b \cdot \nabla b, \\ \partial_{t} b + u \cdot \nabla b + \partial_{22} b = b \cdot \nabla u, \\ \nabla \cdot u = \nabla \cdot b = 0, \\ u (0) = u_{0} (x), b (0) = b_{0} (x) . \end{matrix} \end{matrix}$
Theorem 1.2.
Let $T = [0, 1]$ be a 1 D periodic box and $\bar{Ω} = R \times T$ . Suppose that $(u_{0}, b_{0}) \in H^{2} (R^{2})$ satisfy $\nabla \cdot u_{0} = 0$ , $\nabla \cdot b_{0} = 0$ . Then there exists $δ > 0$ such that, if $\begin{matrix} ‖ u_{0} ‖_{H^{2}} + ‖ b_{0} ‖_{H^{2}} ⩽ δ, \end{matrix}$ then ( 1.3 ) has a unique global solution $(u, b) \in C ([0, \infty); H^{2} (R^{2}))$ satisfying $\begin{matrix} {‖ (u (t), b (t)) ‖}_{H^{2}}^{2} + \int_{0}^{t} {‖ \partial_{2} u (s) ‖}_{H^{2}}^{2} d s + \int_{0}^{t} {‖ \partial_{2} b (s) ‖}_{H^{2}}^{2} d s ⩽ C δ^{2}, \end{matrix}$ for any $t > 0$ and some uniform constant C.
Remark 1.3.
The exponential decay of the corresponding oscillation part of $(u, b)$ can also be further proved, we refer readers to the paper [21].
Remark 1.4.
When $b_{0} = 0,$ the system (1.1) becomes the Navier Stokes equations with the horizontal dissipation. Theorm 1.1 extends the corresponding result in [10].

Motivated by the paper [9], we introduce some notations below. Define the horizontal average, $\begin{matrix} (1.4) & \bar{u} (x_{2}, t) = \int_{T} u (x_{1}, x_{2}, t) d x_{1} . \end{matrix}$ Let $\tilde{u}$ be the corresponding oscillation part $\begin{matrix} (1.5) & \tilde{u} = u - \bar{u} . \end{matrix}$ Similarly, we define $\bar{b}$ , $\tilde{b}$ and $\bar{f}$ , $\tilde{f}$ . A crucial property of $\tilde{u}$ is that it obeys a strong version of the Poincaré type inequality $\begin{matrix} (1.6) & ‖ \tilde{u} ‖_{L^{2} (Ω)} ⩽ C ‖ \partial_{1} \tilde{u} ‖_{L^{2} (Ω)} . \end{matrix}$ Throughout the paper, we denote the norms of usual Lebesgue space $L^{p} (R^{3})$ by $‖ u ‖_{L^{p}}^{p} = \int_{Ω} | u |^{p} d x$ , for $1 ⩽ p < \infty$ , $H^{2} (R^{2})$ is the standard Sobolev space. $C_{i}$ and C denote different positive constants in different place.

The bootstrap argument will be employed in our proof. A rigorous statement of the abstract bootstrap principle can be found in T. Tao’s book (see [26]). The crucial point is to show the following global energy inequality, for any $t > 0$ , $\begin{matrix} (1.7) & E (t) ⩽ C E (0) + C E {(t)}^{\frac{3}{2}}, \end{matrix}$ where $\begin{matrix} E (t) = sup_{0 ⩽ τ ⩽ t} ({‖ u (τ) ‖}_{H^{2}}^{2} + {‖ b (τ) ‖}_{H^{2}}^{2}) + \int_{0}^{t} {‖ \partial_{1} u (s) ‖}_{H^{2}}^{2} d s + \int_{0}^{t} {‖ \partial_{1} b (s) ‖}_{H^{2}}^{2} d s . \end{matrix}$

The paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we prove the key inequality of $E (t)$ . In Section 4 and Section 5, we prove Theorem 1.1 and Theorem 1.2, respectively.
2. Preliminaries

The following lemmas introduced in [9] are useful.

Lemma 2.1.
Assume that the 2D function f defined on $Ω = T \times R$ is sufficiently regular, say $f \in H^{2} (R^{2})$ .
$\bar{f}$ and $\tilde{f}$ obeys the following properties, $\begin{matrix} \overline{\partial_{1} f} = \partial_{1} \bar{f} = 0, \overline{\partial_{2} f} = \partial_{2} \bar{f}, \bar{\tilde{f}} = 0, \tilde{\partial_{2} f} = \partial_{2} \tilde{f} . \end{matrix}$

If f is divergence free vector field, then $\bar{f}$ and $\tilde{f}$ are also divergence free.

$\bar{f}$ and $\tilde{f}$ are orthogonal $L^{2}$ , namely $\begin{matrix} (\bar{f}, \tilde{f}) : = \int_{Ω} \bar{f} \tilde{f} d x = 0, ‖ f ‖_{L^{2} (Ω)}^{2} = ‖ \bar{f} ‖_{L^{2} (Ω)}^{2} + ‖ \tilde{f} ‖_{L^{2} (Ω)}^{2} . \end{matrix}$ In particular, $‖ \bar{f} ‖_{L^{2}} ⩽ ‖ f ‖_{L^{2}}$ and $‖ \tilde{f} ‖_{L^{2}} ⩽ ‖ f ‖_{L^{2}}$ .

Lemma 2.2.
Let $Ω = T \times R$ . For any $f, g, h \in L^{2} (Ω)$ with $\partial_{1} f \in L^{2} (Ω)$ and $\partial_{2} g \in L^{2} (Ω)$ , then $\begin{matrix} | \int_{Ω} \tilde{f} g h d x | ⩽ C ‖ \tilde{f} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{f} ‖_{L^{2}}^{\frac{1}{2}} ‖ g ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} g ‖_{L^{2}}^{\frac{1}{2}} ‖ h ‖_{L^{2}} . \end{matrix}$ For any $f \in H^{2} (Ω)$ , we have $\begin{matrix} ‖ \tilde{f} ‖_{L^{\infty}} ⩽ C ‖ \tilde{f} ‖_{L^{2} (Ω)}^{\frac{1}{4}} ‖ \partial_{1} \tilde{f} ‖_{L^{2} (Ω)}^{\frac{1}{4}} ‖ \partial_{2} \tilde{f} ‖_{L^{2} (Ω)}^{\frac{1}{4}} ‖ \partial_{1} \partial_{2} \tilde{f} ‖_{L^{2} (Ω)}^{\frac{1}{4}} . \end{matrix}$
Lemma 2.3.
Let $\bar{f}$ and $\tilde{f}$ be defined as ( 1.4 ) and ( 1.5 ). If $‖ \partial_{1} \tilde{f} ‖_{L^{2} (Ω)} < \infty$ , then $\begin{matrix} ‖ \tilde{f} ‖_{L^{2} (Ω)} ⩽ C ‖ \partial_{1} \tilde{f} ‖_{L^{2} (Ω)}, \end{matrix}$ where C is a pure constant. In addition, if $‖ \partial_{1} \tilde{f} ‖_{H^{1} (Ω)} < \infty$ , then $\begin{matrix} ‖ \tilde{f} ‖_{L^{\infty} (Ω)} ⩽ C ‖ \partial_{1} \tilde{f} ‖_{H^{1} (Ω)} . \end{matrix}$

3. Global energy inequality

The section is devoted to prove the major estimate in (1.7), namely $\begin{matrix} E (t) ⩽ C E (0) + C E {(t)}^{\frac{3}{2}} . \end{matrix}$

3.1. The $H^{1}$ -estimate

First one has the $L^{2}$ -estimate $\begin{matrix} (3.1) & {‖ (u, b) (t) ‖}_{2}^{2} + 2 \int_{0}^{t} {‖ \partial_{1} u (s) ‖}_{2}^{2} d s + 2 \int_{0}^{t} ‖ \partial_{1} b ‖_{2}^{2} d s = {‖ (u_{0}, b_{0}) ‖}_{2}^{2}, \end{matrix}$ then we give the equations about the vorticity $w = \nabla \times u$ and the current density $j = \nabla \times b$ , $\begin{matrix} (3.2) & \{\begin{matrix} \partial_{t} w + (u \cdot \nabla) w = \partial_{11} w + (b \cdot \nabla) j, \\ \partial_{t} j + (u \cdot \nabla) j = \partial_{11} j + (b \cdot \nabla) w + Q, \end{matrix} \end{matrix}$ where $Q = 2 \partial_{1} b_{1} (\partial_{1} u_{2} + \partial_{2} u_{1}) - 2 \partial_{1} u_{1} (\partial_{1} b_{2} + \partial_{2} b_{1})$ . Taking the $L^{2}$ inner product with $(w, j)$ , one has $\begin{matrix} (3.3) & \begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ (w, j) ‖}_{2}^{2} + ‖ \partial_{1} w ‖_{2}^{2} + ‖ \partial_{1} j ‖_{2}^{2} \\ = \int_{Ω} Q j d x \\ = 2 \int_{Ω} [\partial_{1} b_{1} (\partial_{1} u_{2} + \partial_{2} u_{1}) j - \partial_{1} u_{1} \partial_{1} b_{2} j] d x - 2 \int_{Ω} \partial_{1} u_{1} \partial_{2} b_{1} j d x \\ : = N_{1} + N_{2} + N_{3}, \end{matrix} \end{matrix}$ where we have used the fact $\begin{matrix} \int (b \cdot \nabla) j w d x + \int (b \cdot \nabla) w j d x = 0 . \end{matrix}$ By Hölder inequality, $\begin{matrix} N_{11} & : = 2 \int_{Ω} \partial_{1} b_{1} \partial_{1} u_{2} j d x ⩽ C ‖ \partial_{1} b_{1} ‖_{L^{4}} ‖ \partial_{1} u_{2} ‖_{L^{4}} ‖ j ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}} ‖ \partial_{1} u ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ Note that $\begin{matrix} N_{12} & : = 2 \int_{Ω} \partial_{1} b_{1} \partial_{2} u_{1} j d x = \int \partial_{1} {\tilde{b}}_{1} \partial_{2} {\bar{u}}_{1} \tilde{j} d x + \int \partial_{1} {\tilde{b}}_{1} \partial_{2} {\tilde{u}}_{1} \tilde{j} d x + \int \partial_{1} {\tilde{b}}_{1} \partial_{2} {\tilde{u}}_{1} \bar{j} d x \\ : = N_{121} + N_{122} + N_{123}, \end{matrix}$ where we have use the truth $\begin{matrix} \partial_{1} b_{1} = \partial_{1} ({\bar{b}}_{1} + {\tilde{b}}_{2}) = \partial_{1} {\tilde{b}}_{1}, and \int \partial_{1} {\tilde{b}}_{1} \partial_{2} {\bar{u}}_{1} \bar{j} d x = 0, \end{matrix}$ which can be deduced from Lemma 2.1 and integration by parts. Then by Lemmas 2.1, 2.2 and (1.6), $\begin{array}{l} \begin{matrix} N_{121} & ⩽ C ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}} ‖ \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{2} {\bar{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{2} {\bar{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}}, \end{matrix} \\ \begin{matrix} N_{123} & ⩽ C ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}} ‖ \bar{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}} ‖ \bar{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}} ‖ \partial_{1} u ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix} \end{array}$ $N_{122}$ is similar to the $N_{121}$ or $N_{123}$ . Using the Hölder inequality, it is easy to verify that $\begin{matrix} N_{2} & = \int \partial_{1} u_{1} \partial_{1} b_{2} j d x ⩽ C ‖ \partial_{1} u_{1} ‖_{L^{4}} ‖ \partial_{1} b ‖_{L^{4}} ‖ j ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ Observe that $\begin{matrix} N_{3} & = \int_{Ω} \partial_{1} u_{1} \partial_{2} b_{1} j d x \\ = \int_{Ω} \partial_{1} {\tilde{u}}_{1} \partial_{2} {\bar{b}}_{1} \tilde{j} d x + \int_{Ω} \partial_{1} {\tilde{u}}_{1} \partial_{2} {\tilde{b}}_{1} \bar{j} d x + \int_{Ω} \partial_{1} {\tilde{u}}_{1} \partial_{2} {\tilde{b}}_{1} \tilde{j} d x : = N_{31} + N_{32} + N_{33} . \end{matrix}$ Using Lemmas 2.1, 2.2 and (1.6), one has $\begin{matrix} N_{31} & ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} N_{32} & ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}} ‖ \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \bar{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \bar{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ $N_{33}$ is similar to the $N_{31}$ or $N_{32}$ . So we get $\begin{matrix} (3.4) & {‖ (w, j) (t) ‖}_{2}^{2} + \int_{0}^{t} ‖ \partial_{1} w ‖_{2}^{2} d s + \int_{0}^{t} ‖ \partial_{1} j ‖_{2}^{2} d s ⩽ C E (0) + C E^{\frac{3}{2}} (t) . \end{matrix}$

3.2. The $H^{2}$ -estimate

Taking the inner product of (3.2) with $(- Δ w, - Δ j)$ and integrating by parts, we get $\begin{matrix} (3.5) & \begin{matrix} \frac{1}{2} & \frac{d}{d t} {‖ (\nabla w, \nabla j) ‖}_{2}^{2} + ‖ \partial_{1} \nabla w ‖_{2}^{2} + ‖ \partial_{1} \nabla j ‖_{2}^{2} \\ = \int (u \cdot \nabla) w Δ w d x - \int (b \cdot \nabla) j Δ w d x + \int (u \cdot \nabla) j Δ j d x - \int (b \cdot \nabla) w Δ j d x \\ - 2 \int \partial_{1} b_{1} (\partial_{1} u_{2} + \partial_{2} u_{1}) Δ j d x + 2 \int \partial_{1} u_{1} (\partial_{1} b_{2} + \partial_{2} b_{1}) Δ j d x \\ : = I_{1} + I_{2} + I_{3} + I_{4} + I_{5} + I_{6} . \end{matrix} \end{matrix}$ Now, we estimate these terms respectively. $\begin{matrix} (3.6) & \begin{matrix} I_{1} & = - \int (\nabla u \cdot \nabla w) \nabla w d x \\ = - \int \partial_{1} u_{1} {(\partial_{1} w)}^{2} d x - \int \partial_{1} u_{2} \partial_{1} w \partial_{2} w d x \\ - \int \partial_{2} u_{1} \partial_{1} w \partial_{2} w d x - \int \partial_{2} u_{2} {(\partial_{2} w)}^{2} d x . \end{matrix} \end{matrix}$ Similar to the estimate of the term of N in [9], we have $\begin{matrix} I_{1} ⩽ C ‖ \partial_{1} u ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}} . \end{matrix}$ Note that $\begin{matrix} (3.7) & \begin{matrix} I_{2} + I_{4} & = \int (\nabla b \cdot \nabla) j \nabla w d x + \int (\nabla b \cdot \nabla) w \nabla j d x \\ = \int \partial_{1} b_{1} \partial_{1} j \partial_{1} w d x + \int \partial_{1} b_{2} \partial_{2} j \partial_{1} w d x + \int \partial_{2} b_{1} \partial_{1} j \partial_{2} w d x \\ + 2 \int \partial_{2} b_{2} \partial_{2} j \partial_{2} w d x + \int \partial_{1} b \nabla w \partial_{1} j d x + \int \partial_{2} b_{1} \partial_{1} w \partial_{2} j d x \\ : = K_{1} + K_{2} + K_{3} + K_{4} + K_{5} + K_{6} . \end{matrix} \end{matrix}$ By Hölder inequality, one has $\begin{matrix} K_{1} ⩽ C ‖ \partial_{1} b_{1} ‖_{L^{4}} ‖ \partial_{1} j ‖_{L^{4}} ‖ \partial_{1} w ‖_{L^{2}} ⩽ C ‖ \partial_{1} b ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ u ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} K_{2} ⩽ C ‖ \partial_{1} b_{2} ‖_{L^{4}} ‖ \partial_{1} w ‖_{L^{4}} ‖ \partial_{2} j ‖_{L^{2}} ⩽ C ‖ \partial_{1} b ‖_{H^{2}} ‖ \partial_{1} u ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ To estimate $K_{3}$ , we first write it as $\begin{matrix} K_{3} & = \int \partial_{2} {\bar{b}}_{1} \partial_{1} \tilde{j} \partial_{2} \tilde{w} d x + \int \partial_{2} {\tilde{b}}_{1} \partial_{1} \tilde{j} \partial_{2} \bar{w} d x + \int \partial_{2} {\tilde{b}}_{1} \partial_{1} \tilde{j} \partial_{2} \tilde{w} d x : = K_{31} + K_{32} + K_{33} . \end{matrix}$ From Lemmas 2.1, 2.2 and (1.6), we deduce that $\begin{matrix} K_{31} & ⩽ C ‖ \partial_{1} \tilde{j} ‖_{L^{2}} ‖ \partial_{2} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} \tilde{j} ‖_{L^{2}} ‖ \partial_{1} \partial_{2} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{2} {\bar{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} K_{32} & ⩽ C ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{w} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{w} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}} . \end{matrix}$ $K_{33}$ is similar to the $K_{31}$ or $K_{32}$ .

Now we deal with $K_{4}$ . By $div u = 0$ , one has $\begin{matrix} K_{4} & = \int \partial_{2} b_{2} \partial_{2} j \partial_{2} w d x = - \int \partial_{1} b_{1} \partial_{2} j \partial_{2} w d x \\ = - \int \partial_{1} {\tilde{b}}_{1} \partial_{2} \bar{j} \partial_{2} \tilde{w} d x - \int \partial_{1} {\tilde{b}}_{1} \partial_{2} \tilde{j} \partial_{2} \bar{w} d x - \int \partial_{1} {\tilde{b}}_{1} \partial_{2} \tilde{j} \partial_{2} \tilde{w} d x : = K_{41} + K_{42} + K_{43} . \end{matrix}$ Using Lemmas 2.1, 2.2 and (1.6), there hold $\begin{matrix} K_{41} & ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} K_{42} & ⩽ C ‖ \partial_{2} \bar{w} ‖_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \bar{w} ‖_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}} . \end{matrix}$ $K_{43}$ is similar to the $K_{41}$ or $K_{42}$ . By Hölder inequality, $\begin{matrix} K_{5} ⩽ ‖ \partial_{1} b ‖_{L^{4}} ‖ \partial_{1} j ‖_{L^{4}} ‖ \nabla w ‖_{L^{2}} ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}} . \end{matrix}$ Similarly, we estimate the term $K_{6}$ , $\begin{matrix} K_{6} & = \int \partial_{2} b_{1} \partial_{1} w \partial_{2} j d x \\ = \int \partial_{1} \tilde{w} \partial_{2} {\bar{b}}_{1} \partial_{2} \tilde{j} d x + \int \partial_{1} \tilde{w} \partial_{2} {\tilde{b}}_{1} \partial_{2} \bar{j} d x + \int \partial_{1} \tilde{w} \partial_{2} {\tilde{b}}_{1} \partial_{2} \tilde{j} d x \\ : = K_{61} + K_{62} + K_{63}, \end{matrix}$ moreover, we have $\begin{matrix} K_{61} & ⩽ C ‖ \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} K_{62} & ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{w} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ $K_{63}$ is similar to the $K_{61}$ or $K_{62}$ .

Next we estimate the $I_{3}$ which can be rewrote by $\begin{matrix} (3.8) & \begin{matrix} I_{3} & = \int (\nabla b \cdot \nabla) j \nabla j d x \\ = - \int \partial_{1} u_{1} {(\partial_{1} j)}^{2} d x - \int \partial_{1} u_{2} \partial_{2} j \partial_{1} j d x - \int \partial_{2} u_{1} \partial_{1} j \partial_{2} j d x - \int \partial_{2} u_{2} {(\partial_{2} j)}^{2} d x \\ : = I_{31} + I_{32} + I_{33} + I_{34} . \end{matrix} \end{matrix}$ By Hölder inequality, $\begin{matrix} I_{31} + I_{32} & ⩽ ‖ \partial_{1} u ‖_{L^{4}} ‖ \partial_{1} j ‖_{L^{4}} (‖ \partial_{1} j ‖_{L^{2}} + ‖ \partial_{2} j ‖_{L^{2}}) \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ Noting that $\begin{matrix} I_{33} & = \int \partial_{2} u_{1} \partial_{1} j \partial_{2} j d x = \int \partial_{2} {\bar{u}}_{1} \partial_{1} \tilde{j} \partial_{2} \tilde{j} d x + \int \partial_{2} {\tilde{u}}_{1} \partial_{1} \tilde{j} \partial_{2} \bar{j} d x + \int \partial_{2} {\tilde{u}}_{1} \partial_{1} \tilde{j} \partial_{2} \tilde{j} d x \\ : = I_{331} + I_{332} + I_{333}, \end{matrix}$ one has $\begin{matrix} I_{331} & ⩽ C ‖ \partial_{2} {\bar{u}}_{1} ‖_{L^{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} {\bar{u}}_{1} ‖_{L^{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} I_{332} & ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ \partial_{1} u ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ $I_{333}$ is similar to the $I_{331}$ or $I_{332}$ . It remains to estimate the last term of $I_{3}$ , $\begin{matrix} I_{34} & = \int \partial_{1} u_{1} \partial_{2} j \partial_{2} j d x = \int \partial_{1} {\tilde{u}}_{1} \partial_{2} \bar{j} \partial_{2} \tilde{j} d x + \int \partial_{1} {\tilde{u}}_{1} \partial_{2} \tilde{j} \partial_{2} \tilde{j} d x \\ : = I_{341} + I_{342} . \end{matrix}$ In view of Lemmas 2.1, 2.2 and (1.6), one has $\begin{matrix} I_{341} & ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ $I_{342}$ is similar to the $I_{341}$ .

Now we estimate the $I_{5}$ which can be rewrote by $\begin{matrix} (3.9) & \begin{matrix} I_{5} & = - 2 \int \partial_{1} b_{1} (\partial_{1} u_{2} + \partial_{2} u_{1}) \partial_{1}^{2} j d x - 2 \int \partial_{1} b_{1} (\partial_{1} u_{2} + \partial_{2} u_{1}) \partial_{2}^{2} j d x \\ : = I_{51} + I_{52} . \end{matrix} \end{matrix}$ Using the Hölder inequality and the integration by parts, one gets $\begin{matrix} I_{51} & ⩽ ‖ \partial_{1} b_{1} ‖_{L^{4}} (‖ \partial_{1} u_{2} ‖_{L^{4}} + ‖ \partial_{2} u_{1} ‖_{L^{4}}) {‖ \partial_{1}^{2} j ‖}_{L^{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} I_{52} & = \int \partial_{1} \partial_{2} b_{1} (\partial_{1} u_{2} + \partial_{1} u_{1}) \partial_{2} j d x + \int \partial_{1} b_{1} \partial_{2} \partial_{1} u_{2} \partial_{2} j d x + \int \partial_{1} b_{1} \partial_{2}^{2} u_{1} \partial_{2} j d x \\ : = I_{521} + I_{522} + I_{523}, \end{matrix}$ where $\begin{matrix} I_{521} + I_{522} & ⩽ C (‖ \partial_{1} \partial_{2} b_{1} ‖_{L^{4}} + ‖ \partial_{1} b_{1} ‖_{L^{4}}) (‖ \partial_{1} u_{2} ‖_{L^{4}} + ‖ \partial_{1} u_{1} ‖_{L^{4}} + ‖ \partial_{2} \partial_{1} u_{2} ‖_{L^{4}}) ‖ \partial_{2} j ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}} ‖ \partial_{1} u ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} I_{523} & = \int \partial_{1} b_{1} \partial_{2}^{2} u_{1} \partial_{2} j d x = \int \partial_{1} {\tilde{b}}_{1} \partial_{2}^{2} {\bar{u}}_{1} \partial_{2} \tilde{j} d x + \int \partial_{1} {\tilde{b}}_{1} \partial_{2}^{2} {\tilde{u}}_{1} \partial_{2} \bar{j} d x + \int \partial_{1} {\tilde{b}}_{1} \partial_{2}^{2} {\tilde{u}}_{1} \partial_{2} \tilde{j} d x \\ : = I_{5231} + I_{5232} + I_{5233} . \end{matrix}$ Moreover, there hold $\begin{matrix} I_{5231} & ⩽ C ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{2}^{2} {\bar{u}}_{1} ‖}_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{2}^{2} {\bar{u}}_{1} ‖}_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} b ‖_{H^{2}}^{2} ‖ u ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} I_{5232} & ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{2}^{2} {\tilde{u}}_{1} ‖}_{L^{2}}^{\frac{1}{2}} {‖ \partial_{1} \partial_{2}^{2} {\tilde{u}}_{1} ‖}_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \bar{j} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{1} \partial_{2}^{2} {\tilde{u}}_{1} ‖}_{L^{2}}^{\frac{1}{2}} {‖ \partial_{1} \partial_{2}^{2} {\tilde{u}}_{1} ‖}_{L^{2}} ‖ \partial_{1} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ b ‖_{H^{2}} ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} . \end{matrix}$ $I_{5233}$ is similar to the $I_{5231}$ or $I_{5232}$ . To complete the estimate of $I_{6}$ , we first write it by $\begin{matrix} (3.10) & \begin{matrix} I_{6} & = 2 \int \partial_{1} u_{1} (\partial_{1} b_{2} + \partial_{2} b_{1}) \partial_{1}^{2} j d x + 2 \int \partial_{1} u_{1} (\partial_{1} b_{2} + \partial_{2} b_{1}) \partial_{2}^{2} j d x \\ : = I_{61} + I_{62} . \end{matrix} \end{matrix}$ Then by Hölder inequality and the integration by parts, we have $\begin{matrix} I_{61} & ⩽ ‖ \partial_{1} u_{1} ‖_{L^{4}} (‖ \partial_{1} b_{2} ‖_{L^{4}} + ‖ \partial_{2} b_{1} ‖_{L^{4}}) {‖ \partial_{1}^{2} j ‖}_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ b ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} I_{62} & = 2 \int \partial_{2} j \partial_{2} \partial_{1} u_{1} \partial_{1} b_{2} d x + 2 \int \partial_{2} j \partial_{2} \partial_{1} u_{1} \partial_{2} b_{1} d x \\ + 2 \int \partial_{2} j \partial_{1} u_{1} \partial_{2} \partial_{1} b_{2} d x + 2 \int \partial_{2} j \partial_{1} u_{1} \partial_{2}^{2} b_{1} d x \\ : = M_{1} + M_{2} + M_{3} + M_{4} . \end{matrix}$ It is easy to verify that $\begin{matrix} M_{1} + M_{3} & ⩽ (‖ \partial_{2} \partial_{1} u_{1} ‖_{L^{4}} + ‖ \partial_{1} u_{1} ‖_{L^{4}}) (‖ \partial_{1} b_{2} ‖_{L^{4}} + ‖ \partial_{2} \partial_{1} b_{2} ‖_{L^{4}}) ‖ \partial_{2} j ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ Note that $\begin{matrix} M_{2} & = \int \partial_{2} j \partial_{2} \partial_{1} u_{1} \partial_{2} b_{1} d x \\ = \int \partial_{2} \bar{j} \partial_{2} \partial_{1} {\tilde{u}}_{1} \partial_{2} {\tilde{b}}_{1} d x + \int \partial_{2} \tilde{j} \partial_{2} \partial_{1} {\tilde{u}}_{1} \partial_{2} {\bar{b}}_{1} d x + \int \partial_{2} \tilde{j} \partial_{2} \partial_{1} {\tilde{u}}_{1} \partial_{2} {\tilde{b}}_{1} d x \\ : = M_{21} + M_{22} + M_{23} . \end{matrix}$ It follows from Lemmas 2.1, 2.2 and (1.6) that $\begin{matrix} M_{21} & ⩽ C ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{b}}_{1} ‖_{L^{2}}^{\frac{1}{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} M_{22} & ⩽ C ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}} \\ ⩽ C ‖ \partial_{2} \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} {\bar{b}}_{1} ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ $M_{23}$ is similar to the $M_{21}$ or $M_{22}$ .

To complete the proof, we deal with the last term $M_{4}$ . First we write it as $\begin{matrix} M_{4} & = \int \partial_{2} j \partial_{1} u_{1} \partial_{2}^{2} b_{1} d x \\ = \int \partial_{2} \bar{j} \partial_{1} {\tilde{u}}_{1} \partial_{2}^{2} {\tilde{b}}_{1} d x + \int \partial_{2} \tilde{j} \partial_{1} {\tilde{u}}_{1} \partial_{2}^{2} {\bar{b}}_{1} d x + \int \partial_{2} \tilde{j} \partial_{1} {\tilde{u}}_{1} \partial_{2}^{2} {\tilde{b}}_{1} d x \\ : = M_{41} + M_{42} + M_{43} . \end{matrix}$ Applying Lemmas 2.1, 2.2 and (1.6) again, one has $\begin{matrix} M_{41} & ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{2}^{2} {\tilde{b}}_{1} ‖}_{L^{2}}^{\frac{1}{2}} {‖ \partial_{1} \partial_{2}^{2} {\tilde{b}}_{1} ‖}_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{1} \partial_{2}^{2} {\tilde{b}}_{1} ‖}_{L^{2}}^{\frac{1}{2}} {‖ \partial_{1} \partial_{2}^{2} {\tilde{b}}_{1} ‖}_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \bar{j} ‖_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}}, \end{matrix}$ and $\begin{matrix} M_{42} & ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{2}^{2} {\bar{b}}_{1} ‖}_{L^{2}} \\ ⩽ C ‖ \partial_{1} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} {\tilde{u}}_{1} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{1} \partial_{2} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} ‖ \partial_{2} \partial_{1} \tilde{j} ‖_{L^{2}}^{\frac{1}{2}} {‖ \partial_{2}^{2} {\bar{b}}_{1} ‖}_{L^{2}} \\ ⩽ C ‖ \partial_{1} u ‖_{H^{2}} ‖ \partial_{1} b ‖_{H^{2}} ‖ b ‖_{H^{2}} . \end{matrix}$ $M_{43}$ is similar to the $M_{41}$ or $M_{42}$ .

Combining with all above estimates, we get $\begin{matrix} (3.11) & \begin{matrix} {‖ (\nabla w, \nabla j) (t) ‖}_{2}^{2} + \int_{0}^{t} ‖ \partial_{1} \nabla w ‖_{2}^{2} d s + \int_{0}^{t} ‖ \partial_{1} \nabla j ‖_{2}^{2} d s \\ ⩽ C E (0) + C E^{\frac{3}{2}} (t), \end{matrix} \end{matrix}$ which, together with (3.1) and (3.4), yields $\begin{matrix} E (t) ⩽ C E (0) + C E^{\frac{3}{2}} (t) . \end{matrix}$

4. Proof of Theorem 1.1

Generally speaking, the bootstrap argument starts with an assumption that $E (t)$ is bounded, $\begin{matrix} E (t) ⩽ M, \end{matrix}$ and shows that $E (t)$ actually admits a smaller bounded, say $\begin{matrix} E (t) ⩽ \frac{1}{2} M, \end{matrix}$ provided that the initial $E (0)$ is sufficiently small.

To proceed the bootstrapping argument, we assume that $\begin{matrix} M ⩽ \frac{1}{4 C^{2}} . \end{matrix}$ By (1.7), we get $\begin{matrix} (4.1) & \begin{matrix} E (t) (1 - C E^{\frac{1}{2}} (t)) & ⩽ C E (0), \end{matrix} \end{matrix}$ thus $\begin{matrix} E (t) ⩽ 2 C E (0) . \end{matrix}$ If we choose $E (0) ⩽ \frac{M}{4 C}$ , we deduce that $\begin{matrix} E (t) ⩽ \frac{1}{2} M . \end{matrix}$ Therefore we complete the proof of Theorem 1.1.

5. Proof of Theorem 1.2

Although the theorem can be proven in a similar method as that of Theorem 1.1, motivated by [6], we provide a more concise proof.

Set $\begin{array}{l} (5.1) & \begin{array}{l} U_{1} (x, y, t) = u_{2} (y, x, t), U_{2} (x, y, t) = u_{1} (y, x, t), P (x, y, t) = p (y, x, t) \\ B_{1} (x, y, t) = b_{2} (y, x, t), B_{2} (x, y, t) = b_{1} (y, x, t) . \end{array} \end{array}$ Obviously, the $(x, y)$ of $U (x, y) : = (U_{1} (x, y), U_{2} (x, y))$ belongs to domain $Ω = T \times R$ .

Then $U (x, y)$ , P and $B (x, y) : = (B_{1} (x, y), B_{2} (x, y))$ satisfy $\begin{matrix} \{\begin{matrix} \partial_{t} U + U \cdot \nabla U + \partial_{11} U + \nabla P = b \cdot \nabla B, \\ \partial_{t} B + U \cdot \nabla B + \partial_{11} B = B \cdot \nabla U, \\ \nabla \cdot U = \nabla \cdot B = 0, \\ U (0) = U_{0} (x), B (0) = B_{0} (x), \end{matrix} \end{matrix}$ where we have used the equality $\begin{matrix} U_{1} \partial_{1} U_{1} + U_{2} \partial_{2} U_{1} = u_{2} (y, x, t) \partial_{2} u_{2} (y, x, t) + u_{1} (y, x, t) \partial_{1} u_{2} (y, x, t) . \end{matrix}$ Note that $\begin{array}{l} (5.2) & \begin{array}{l} ‖ U ‖_{H^{2} (Ω)} = ‖ u ‖_{H^{2} (\bar{Ω})}, ‖ B ‖_{H^{2} (Ω)} = ‖ b ‖_{H^{2} (\bar{Ω})} . \\ ‖ \partial_{11} U ‖_{H^{2} (Ω)} = ‖ \partial_{22} u ‖_{H^{2} (\bar{Ω})}, ‖ \partial_{11} B ‖_{H^{2} (Ω)} = ‖ \partial_{22} b ‖_{H^{2} (\bar{Ω})} . \end{array} \end{array}$ Indeed, there holds $\begin{array}{l} \int_{[0, 1] \times R} {| u_{2} (y, x, t) |}^{2} d x d y = \int_{R \times [0, 1]} {| u_{2} (x, y, t) |}^{2} d x d y, \\ \partial_{x} (U_{1} (x, y)) = \partial_{x} (u_{2} (y, x)) = \partial_{y} u_{2} (y, x), \\ \partial_{x} \partial_{y} (U_{1} (x, y)) = \partial_{x} \partial_{y} (u_{2} (y, x)) = \partial_{x} \partial_{y} u_{2} (y, x), \end{array}$ and $\begin{matrix} \partial_{x x} (U_{1} (x, y)) = \partial_{x x} (u_{2} (y, x)) = \partial_{y} \partial_{y} u_{2} (y, x) . \end{matrix}$ Then we obtain $\begin{matrix} sup_{0 ⩽ τ ⩽ t} ({‖ u (τ) ‖}_{H^{2} (\bar{Ω})}^{2} + {‖ b (τ) ‖}_{H^{2} (\bar{Ω})}^{2}) + \int_{0}^{t} {‖ \partial_{2} u (s) ‖}_{H^{2} (\bar{Ω})}^{2} d s + \int_{0}^{t} {‖ \partial_{2} b (s) ‖}_{H^{2} (\bar{Ω})}^{2} d s \\ = sup_{0 ⩽ τ ⩽ t} ({‖ U (τ) ‖}_{H^{2} (Ω)}^{2} + {‖ B (τ) ‖}_{H^{2} (Ω)}^{2}) + \int_{0}^{t} {‖ \partial_{1} U (s) ‖}_{H^{2} (Ω)}^{2} d s + \int_{0}^{t} {‖ \partial_{1} B (s) ‖}_{H^{2} (Ω)}^{2} d s . \end{matrix}$ Thus we can prove the Theorem 1.2 by the result of Theorem 1.1.

Footnotes

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Nos. 11801574, 11971485, 12171486), Natural Science Foundation of Hunan Province (No. 2019JJ50788), Central South University Innovation-Driven Project for Young Scholars (No. 2019CX022) and Fundamental Research Funds for the Central Universities of Central South University, China (Nos. 2020zzts038, 2021zzts0041).

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Stability for the 2D MHD equations with horizontal dissipation

Abstract

Keywords

1. Introduction and main results

3.1. The H 1 -estimate

3.2. The H 2 -estimate

4. Proof of Theorem 1.1

5. Proof of Theorem 1.2

Footnotes

Acknowledgements

References

3.1. The $H^{1}$ -estimate

3.2. The $H^{2}$ -estimate