1. Introduction

Let $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a one-dimensional (1D) periodic domain and $\mathbb {R}$ being the real line. Consider the 2D incompressible magnetohydrodynamic (MHD) equations with horizontal dissipation

(1.1)\begin{equation} \begin{cases} \partial_t u + u\cdot\nabla u ={-}\nabla P + \nu \, \partial _{11} u + B\cdot\nabla B, \quad x \in \Omega,\quad t>0, \\ \partial_t B+ u\cdot\nabla B + \eta B= B\cdot\nabla u, \\ \nabla\cdot u=0, \quad \nabla \cdot B=0, \\ u(x,0) =u_0(x), \quad B(x,0)=B_0(x), \end{cases} \end{equation}

where $u$ denotes the velocity field, $B$ the magnetic field and $P$ the pressure, and $\nu >0$ and $\eta$ are the viscosity and the damping coefficients, respectively. Here the velocity $u$ obeys a degenerate Navier–Stokes equation with only horizontal dissipation $\nu \partial _{11}u$ and with a Lorentz forcing term. The magnetic field $B$ satisfies the induction equation with a damping term. The goal of this paper is to understand the stability and the large-time behaviour of perturbations near a background magnetic field.

This study is partially motivated by the stabilizing phenomenon of a background magnetic field on electrically conducting fluids that has been observed in physical experiments and numerical simulations (see, e.g., [Reference Alexakis1, Reference Alfvén2, Reference Burattini, Zikanov and Knaepen6, Reference Davidson12–Reference Davidson14, Reference Gallet, Berhanu and Mordant24, Reference Gallet and Doering25]). Since the dynamics of electrically conducting fluids is governed by the MHD equations (see, e.g., [Reference Biskamp4, Reference Priest and Forbes39]), the aim here is to establish this remarkable observation as a mathematically rigorous fact on the MHD equations.

We take the background magnetic field to be the unit vector in the $x_1$-direction, $B^{(0)}=(1,\,0)$. The corresponding steady-state solution of (1.1) is given by

\[ u^{(0)} =(0,0), \quad B^{(0)}=(1,0). \]

We write $(u,\, b)$ with $b= B- B^{(0)}$ for the perturbation near $(u^{(0)},\, B^{(0)})$. Our attention will be focussed on the following new system

(1.2)\begin{equation} \begin{cases} \partial _t u+ u\cdot \nabla u + \nabla P = \nu \partial_{11}u +b\cdot \nabla b +\partial_{1} b, \quad\ x\in \Omega,\quad t>0\\ \partial_t b + u \cdot \nabla b + \eta b = b\cdot \nabla u+ \partial_1{u} , \\ \nabla \cdot u=\nabla \cdot b=0, \\ u(x, 0)=u_0(x), ~b(x, 0)=b_0(x). \end{cases} \end{equation}

In comparison with the original system in (1.1), there are two extra terms $\partial _1b$ and $\partial _1 u$ in (1.2). We aim to achieve a complete understanding on the stability of solutions to (1.2) in the Sobolev setting. In addition, we also attempt to obtain the precise large-time behaviour of $(u,\, b)$ and establish the eventual dynamics of (1.2).

Due to the lack of vertical dissipation in (1.2), the resolution of the stability problem is not direct. If we follow the standard energy method approach, the difficulty is immediate. The divergence-free conditions $\nabla \cdot u=\nabla \cdot b=0$ allow us to obtain a suitable upper bound on the $H^{1}$-norm of $(u,\, b)$, but it does not appear to be possible to control the $H^{2}$-norm directly. Even if we completely ignore the terms related to the magnetic field and simply consider the 2D anisotropic Navier–Stokes equations

\[ \partial _t u+ u\cdot \nabla u + \nabla P = \nu \partial_{11}u, \]

direct energy estimates fail to generate a suitable $H^{2}$-bound. In fact, when we resort to the corresponding vorticity formulation

\[ \partial_t \omega + u\cdot\nabla \omega = \nu \partial_{11} \omega, \]

the one-directional dissipation is insufficient to bound the nonlinearity directly. In the estimate of $\nabla \omega$,

\[ \frac 12 \frac{d}{dt} \|\nabla \omega\|_{L^{2}}^{2} + \nu \|\partial_1\nabla \omega\|_{L^{2}}^{2} ={-} \int \nabla \omega \cdot \nabla u\cdot \nabla \omega\,dx, \]

the right-hand side does not admit a suitable upper bound. In fact,

(1.3)\begin{align} \int \nabla \omega \cdot \nabla u\cdot \nabla \omega\,dx & = \int \partial_1 u_1 (\partial_1 \omega)^{2}\,dx + \int \partial_1 u_2 \partial_1 \omega \partial_2\omega\,dx \nonumber\\ & \quad + \int \partial_2 u_1 \partial_1 \omega\, \partial_2\omega\,dx + \int \partial_2 u_2 (\partial_2\omega)^{2}\,dx \end{align}

and the two terms in (1.3) can not be controlled suitably.

One novel idea to overcome this difficulty is to explore the stabilizing effect of the magnetic field on the fluids as hinted by the aforementioned experimental results. Mathematically we make full use of the coupling and interaction in the MHD system in (1.2) to unearth the hidden smoothing and stabilizing properties. To do so, we first apply the Leray projection $\mathbb {P} = I -\nabla \Delta ^{-1}\nabla \cdot$ to the velocity equation to eliminate the pressure,

\[ \partial_t u = \nu \partial_{11} u + \partial_1 b + {\mathbb{P}}(b\cdot\nabla b- u\cdot\nabla u). \]

By differentiating the linearized system in time

(1.4)\begin{equation} \begin{cases} \partial_t u = \nu \partial_{11} u + \partial_1 b, \\ \partial_t b ={-}\eta b + \partial_1 u \end{cases} \end{equation}

and making several substitutions, we can convert (1.4) into a system of wave equations

(1.5)\begin{equation} \begin{cases} \partial_{tt} u + (\eta + \nu \partial_{11})\partial_t u - (1+ \nu \eta) \partial_{11} u=0, \\ \partial_{tt} b + (\eta + \nu \partial_{11})\partial_t b - (1+ \nu \eta) \partial_{11} b=0. \end{cases} \end{equation}

(1.5) allows us to decouple $u$ and $b$ and exhibits more smoothing and stabilizing properties than (1.4). In particular, both $u$ and $b$ gain weak horizontal dissipation as can be seen from the pieces $(1+ \nu \eta ) \partial _{11} u$ and $(1+ \nu \eta ) \partial _{11} b$. Unfortunately, this extra regularization does not appear to help with the deficiency of vertical dissipation in the velocity equation. As a consequence, this approach fails.

We remark that a previous work of Feng, Hafeez and Wu [Reference F. Hafeez and Wu23] explored the extra stabilizing and smoothing of the wave structure, and successfully resolved the stability problem on the same MHD system near the background magnetic field $B^{(0)} =(0,\,1)$. When the background magnetic field is $(0,\,1)$, the extra regularity is in the vertical direction and complements with the horizontal dissipation in the velocity equation. Therefore, the direction of the background magnetic field plays a crucial role in the stabilizing phenomenon on electrically conduction fluids.

This paper seeks a different approach to resolve the stability problem concerned here. The spatial domain here is $\Omega = \mathbb {T}\times \mathbb {R}$ and we take full advantage of the geometry of this domain. The horizontal direction is periodic and we can separate the zeroth Fourier mode from the non-zero ones. The zeroth Fourier mode corresponds to the horizontal average. This hints the decomposition of the physical quantities into the horizontal averages and the corresponding oscillation parts. More precisely, for a function $f$ that is integrable in $x\in \mathbb {T}$, we define

\[ \bar f(x_2) = \int_{\mathbb{T}} f(x_1, x_2) \,dx_1, \quad f = \bar f + \widetilde f. \]

This decomposition is orthogonal in the Sobolev space $H^{k}(\Omega )$ for any integer $k\ge 0$ (see lemma 2.2 in § 2). More crucially, the oscillation part $\widetilde f$ obeys a strong version of the Poincaré type inequality

\[ \|\widetilde f\|_{L^{2}(\Omega)} \le C\, \|\partial_1 \widetilde f\|_{L^{2}(\Omega)}. \]

This inequality allows us to control some of the nonlinear parts in terms of the horizontal dissipation. By invoking the decompositions

\[ u = \bar u + \widetilde u, \quad b = \bar b + \widetilde b \]

and applying the aforementioned Poincaré inequality together with various anisotropic inequalities, we are able to successfully bound the nonlinearity and establish the following stability result.

Theorem 1.1 Let $\eta >0$ and $\nu >0$. Consider (1.2) with the initial data $(u_0,\, b_0) \in H^{3}(\Omega ),$ and $\nabla \cdot u_0= \nabla \cdot b_0=0$. Then there exists a constant $\varepsilon =\varepsilon (\nu,\, \eta )>0$ such that, if

\[ \|u_0\|_{H^{3}}+ \|b_0\|_{H^{3}} \leq \varepsilon, \]

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

\[ \|u(t)\|^{2}_{H^{3}}+ \|b(t)\|_{H^{3}}^{2}+\displaystyle\int_0^{t} \left(\|\partial_1 u\|_{H^{3}}^{2} + \|b\|_{H^{3}}^{2}\right) ~d\tau \le C\, \varepsilon^{2}, \]

where $C>0$ is independent of $\varepsilon$ and $t$.

Theorem 1.1 successfully resolves the stability problem on a partially dissipated MHD system near a background magnetic field even when the smoothing effect of the magnetic field is not sufficient to deal with the dissipation deficiency.

Efforts are also devoted to understanding the precise large-time behaviour of the perturbation. We expect the horizontal average $(\bar u,\, \bar b)$ to behave differently from the oscillation part $(\widetilde u,\, \widetilde b)$. Intuitively $(\bar u,\, \bar b)$ corresponds to the zeroth horizontal Fourier mode and the associated dissipation term vanishes. Thus $(\bar u,\, \bar b)$ may not decay in time. In contrast, $(\widetilde u,\, \widetilde b)$ consists of non-zero horizontal Fourier modes and the horizontal dissipation effectively plays the role of damping. As a consequence, $(\widetilde u,\, \widetilde b)$ could decay exponentially in time. Our second theorem rigorously confirms this intuition.

Theorem 1.2 Let $u_0,\, b_0\in H^{3}(\Omega )$ with $\nabla \cdot u_0=0$ and $\nabla \cdot b_0=0$. Assume that $\|u_0\|_{H^{3}}+ \|b_0\|_{H^{3}} \leq \varepsilon$ for sufficiently small $\varepsilon >0$. Let $(u,\, b)$ be the corresponding solution of (1.2). Then the $H^{1}$ norm of the oscillation part $(\widetilde u,\, \widetilde b)$ decays exponentially in time,

(1.6)\begin{equation} \|\widetilde{u}(t)\|_{H^{1}}+\|\widetilde{b}(t)\|_{H^{1}} \leq (\|u_0\|_{H^{1}}+\|b_0\|_{H^{1}}) e^{{-}C_1t}, \end{equation}

for some constant $C_1>0$ and for all $t>0$.

We explain the main lines in the proof of theorem 1.1. The local well-posedness of (1.2) in the Sobolev space $H^{3}(\Omega )$ can be shown via standard procedures such as the approach in the book of Majda and Bertozzi [Reference Majda and Bertozzi37]. Our attention is focussed on the global bound of $(u,\, b)$ in $H^{3}(\Omega )$. One of the most suitable tools for this purpose is the bootstrapping argument [Reference Tao45]. To set up the argument, we first construct the energy functional. For the MHD system in (1.2), the energy functional $E(t)$ is naturally given by the $H^{3}$-norm of $(u,\,b)$ together with the time integrals from dissipative and damping terms, namely

\[ E(t)= \displaystyle\sup_{0 \leq \tau \leq t} \{ \|u(\tau)\|_{H^{3}}^{2}+ \|b(\tau)\|_{H^{3}}^{2} \}+ 2\nu\displaystyle\int_0^{t} \|\partial_1 u\|_{H^{3}}^{2}~d\tau + 2\eta \displaystyle\int_0^{t} \|b\|_{H^{3}}^{2} ~d\tau. \]

The main effort is then devoted to proving the energy inequality

(1.7)\begin{equation} E(t)\leq E(0)+ \ C E^{\frac{3}{2}}(t). \end{equation}

Once (1.7) is at our disposal, the bootstrapping argument then implies that, if $E(0) :=\|(u_0,\, b_0)\|_{H^{3}}^{2}$ is sufficiently small, say

\[ \|(u_0, b_0)\|_{H^{3}} \le \varepsilon \]

for some suitable $\varepsilon >0$, then $E(t)$ remains uniformly bounded for any $t>0$,

\[ E(t) \le C\, \varepsilon^{2}, \]

which gives us the desired global bound on $\|(u(t),\, b(t))\|_{H^{3}}$. To prove (1.7), we invoke the orthogonal decompositions $u = \bar u+ \widetilde u$ and $b = \bar b + \widetilde b$, apply the Poincaré type inequalities and anisotropic upper bounds for triple products. More technical details are provided in § 3.

To prove theorem 1.2, we first take the horizontal average of (1.2) to obtain the equations of $(\bar u,\, \bar b)$,

(1.8)\begin{equation} \begin{cases} \partial _t \bar{u}+ \overline{u\cdot \nabla \widetilde{u}} + \left(\begin{array}{c} 0 \\ \partial_2\bar{p}\end{array} \right) = \overline{b\cdot \nabla \widetilde{b} },\\ \partial_t \bar{b} + \overline{u \cdot \nabla \widetilde{b} }+ \eta \bar{b} = \overline{b\cdot \nabla \widetilde{u}}. \end{cases} \end{equation}

We then write the equations of $(\widetilde u,\, \widetilde b)$ by taking the difference of (1.2) and (1.8),

(1.9)\begin{equation} \begin{cases} \partial_t \widetilde{u}+ \widetilde{u\cdot \nabla \widetilde{u}}+u_2\partial_2\bar{u}+\nabla \widetilde{p}-\nu \partial_1^{2} \widetilde{u}-\widetilde{b\cdot \nabla \widetilde{b}}-b_2\partial_2\bar{b}-\partial_1\widetilde{b}=0,\\ \partial_t \widetilde{b} + \widetilde{u\cdot \nabla \widetilde{b}}+u_2\partial_2\bar{b}+\eta\widetilde{b}-\widetilde{b\cdot \nabla \widetilde{u}}-b_2\partial_2\bar{u}-\partial_1\widetilde{u}=0. \end{cases} \end{equation}

The proof of (1.6) is divided into the estimates of $\|(\widetilde u,\, \widetilde b)\|_{L^{2}}$ and $\|(\nabla \widetilde u,\, \nabla \widetilde b)\|_{L^{2}}$. The efforts are devoted to bounding the nonlinearity in terms of the horizontal derivatives of $\widetilde u$. Poincaré's inequality and anisotropic upper bounds for the triple products are used extensively. After a tedious process of evaluating many terms, we obtain

\begin{align*} & \dfrac{d}{dt} (\|\widetilde{u}\|_{H^{1}}^{2}+ \|\widetilde{b}\|_{H^{1}}^{2}) +(2\nu -C\|(u, b)\|_{H^{3}}) \|\partial_1\widetilde{u}\|_{H^{1}}^{2} \\ & \quad +(2\eta-C\|(u, b)\|_{H^{3}})\|\widetilde{b}\|_{H^{1}}\leq 0, \end{align*}

which yields the decay rate in (1.6). A detailed proof is provided in § 4.

Finally we briefly summarize some of related results to provide a broader view on the studies of the MHD equations. Fundamental issues on the MHD equations such as well-posedness and stability problems have attracted a lot of attention. Substantial progress has recently been made on the well-posedness problem concerning the MHD equations with various partial or fractional dissipation (see, e.g., [Reference Cao, Regmi and Wu8–Reference Cao, Wu and Yuan10, Reference Dong, Jia, Li and Wu16, Reference Dong, Li and Wu17, Reference Du and Zhou19–Reference Fefferman, McCormick, Robinson and Rodrigo22, Reference Jiu, Niu, Wu, Xu and Yu30, Reference Li, Tan and Yin32, Reference Lin and Du35, Reference Schonbek, Schonbek and Süli42, Reference Sermange and Temam43, Reference Wan46, Reference Wu49, Reference Xiao, Xin and Wu53, Reference Yamazaki55–Reference Yuan and Zhao61]). Since the pioneering work of Alfvén [Reference Alfvén2], the stability problem on various MHD systems has recently gained renewed interests and there are substantial developments. By taking advantage of the Elsässer variables, several papers have successfully solved the stability problem on the ideal MHD equations or the fully dissipated MHD equations with identical (or almost identical) viscosity and magnetic diffusivity (see [Reference Bardos, Sulem and Sulem3, Reference Cai and Lei7, Reference He, Xu and Yu26, Reference Wei and Zhang47]). The stability problem on the MHD equations with only kinematic dissipation in $\mathbb {R}^{2}$ or $\mathbb {R}^{3}$ have been solved via different approaches [Reference Deng and Zhang15, Reference Hu27, Reference Hu and Lin28, Reference Lin, Xu and Zhang33, Reference Lin and Zhang34, Reference Ren, Wu, Xiang and Zhang40, Reference Ren, Xiang and Zhang41, Reference Tan and Wang44, Reference Wu and Wu50, Reference Wu, Wu and Xu51, Reference Zhang62, Reference Zhang63]. The same problem in the periodic setting $\mathbb {T}^{3}$ has been investigated by [Reference Pan, Zhou and Zhu38]. The MHD equations with only magnetic diffusivity have recently been studied for the small data global well-posedness near the trivial solution or a background magnetic field [Reference Chen, Zhang and Zhou11, Reference Ji and Wu29, Reference Wei and Zhang48, Reference Xu and Zhang54, Reference Zhou and Zhu64], although a complete solution on the stability problem near a background magnetic field is currently lacking. When the velocity equation involves only horizontal or vertical dissipation, the velocity equation itself alone may not be stable and the stability problem relies on the enhanced dissipation resulting from the coupling and interaction. Several such MHD systems with degenerate velocity dissipation have been shown to be stable near suitable background magnetic fields [Reference Boardman, Lin and Wu5, Reference F. Hafeez and Wu23, Reference Li, Wu and Xu31, Reference Lin, Ji, Wu and Yan36, Reference Wu and Zhu52].

The rest of this paper is divided into three sections. Section 2 states several properties on the aforementioned decomposition and provides several anisotropic inequalities. Section 3 proves theorem 1.1 while § 4 presents the proof of theorem 1.2.