1. Introduction

This paper studies the stability properties of the two-dimensional incompressible magnetohydrodynamic (MHD) system with only vertical dissipation, given by

(1.1)\begin{equation} \left\{ \begin{aligned} &\partial_{t}U+U\cdot\nabla U-\nu\partial_{yy} U+\nabla P=B\cdot\nabla B, \\[4pt] &\partial_{t}B+U\cdot\nabla B-\mu\partial_{yy} B=B\cdot\nabla U,\\[4pt] &{\rm div}\, U={\rm div}\, B=0, \end{aligned} \right. \end{equation}

posed on the periodic strip $\Omega:=\mathbb{T}\times \mathbb{R}$, where $U=(U_1,U_2)$ and $B=(B_1,B_2)$ denote the velocity and magnetic field, respectively, and $P$ the pressure. The positive constants $\nu$ and $\mu$ represent the viscosity and magnetic diffusivity coefficients.

Anisotropic dissipation arises naturally in many physical systems, ranging from atmospheric circulation to plasma dynamics (see, e.g., [Reference Cao, Li and Titi4, Reference Cao and Wu5, Reference Elgindi and Widmayer13, Reference Ji, Tian and Wu17, Reference Paicu and Zhang31, Reference Paicu and Zhang32, Reference Xu and Zhang41]). In particular, geophysical fluid dynamics provides a canonical example: turbulent diffusivity in the vertical direction is often orders of magnitude larger than in the horizontal direction. The absence of full dissipation renders the classical energy method insufficient, requiring refined tools such as frequency-space decompositions and anisotropic energy estimates. This intermediate regime, situated between full dissipation and nondissipative dynamics, plays a crucial role in understanding incomplete turbulence observed in natural systems.

Since Reynolds’s pioneering experiments [Reference Reynolds36], understanding the transition from laminar to turbulent flows has remained a central challenge in fluid dynamics. Kelvin’s classical analysis in 1887 [Reference Kelvin19] already revealed that laminar flows may persist at large Reynolds numbers if the initial perturbation is sufficiently small. A more extensive physical discussion can be found in [Reference Chandrasekhar6, Reference Davidson9, Reference Hughes and Tobias16]. Over the past decade, remarkable progress has been made towards rigorously establishing stability thresholds for Couette flows in various fluid models, giving rise to a rapidly growing literature (see, e.g., [Reference Bedrossian, Germain and Masmoudi1–Reference Bedrossian, Vicol and Wang3, Reference Chen, Li, Wei and Zhang7, Reference Chen, Wei and Zhang8, Reference Deng, Wu and Zhang10–Reference Dolce, Knobel and Zillinger12, Reference Huang, Sun and Xu14, Reference Huang, Sun and Xu15, Reference Jin, Ren and Wei18, Reference Knobel and Zillinger22–Reference Niu, Wu, Wu and Xu30, Reference Rao, Zhang and Zi33–Reference Ren and Wei35, Reference Wang and Zhang37–Reference Wu, Wu and Xu40, Reference Zeng, Zhang and Zi42, Reference Zhao and Zi43]).

It is clear that the Couette flow

\begin{equation*} \tilde{U}=(y,0), \qquad \tilde{B}=(0,0) \end{equation*}

is a stationary solution of (1.1). Our aim is to analyse the nonlinear stability of this flow within the framework of 2D MHD with vertical dissipation. Introducing perturbations

\begin{equation*} u=U-\tilde{U}, \qquad \mathfrak{B}=B-\tilde{B}, \end{equation*}

the dynamics of $(u,\mathfrak{B})$ is governed by

(1.2)\begin{equation} \left\{ \begin{aligned} &\partial_{t}u+y\partial_{x}u-\nu\partial_{yy}u+(u_{2},0)^{t}=-\nabla p-u\cdot\nabla u+\mathfrak{B}\cdot \nabla \mathfrak{B},\\[4pt] &\partial_{t}\mathfrak{B}+y\partial_{x}\mathfrak{B}-\mu\partial_{yy}\mathfrak{B}-(\mathfrak{B}_{2},0)^{t}=-u\cdot\nabla \mathfrak{B}+\mathfrak{B}\cdot\nabla u,\\[4pt] &{\rm div}\, u={\rm div}\, \mathfrak{B}=0,\\[4pt] &(u,\mathfrak{B})(t,x,y)|_{t=0}=(u_{0},\mathfrak{B}_{0})(x,y), \end{aligned} \right. \end{equation}

where the pressure is determined by

\begin{equation*} p=(-\Delta)^{-1}\!\left(2\partial_{x}u_{2}\right)+(-\Delta)^{-1}\nabla\cdot(u\cdot\nabla u-\mathfrak{B}\cdot\nabla \mathfrak{B}). \end{equation*}

To simplify the equations in (1.2), we introduce the vorticity $\omega=\nabla\times u$ and the magnetic stream function $\phi$ defined by $\phi=\Delta^{-1}j$ with current density $j=\nabla\times \mathfrak{B}$. In terms of $(\omega,\phi)$, the system takes the form

(1.3)\begin{equation} \left\{ \begin{aligned} &\partial_{t}\omega+y\partial_{x}\omega-\nu\partial_{yy}\omega=-u\cdot\nabla\omega+\mathfrak{B}\cdot\nabla\Delta\phi,\\[4pt] &\partial_{t}\phi+y\partial_{x}\phi-\mu\partial_{yy}\phi=-u\cdot\nabla\phi,\\[4pt] &u=-\nabla^{\perp}(-\Delta)^{-1}\omega, \qquad \mathfrak{B}=\nabla^{\perp}\phi. \end{aligned} \right. \end{equation}

The focus of this work is precisely on the vertical dissipation regime, where dissipation acts exclusively in the $y$-direction. Compared to prior works, our analysis does not rely on coordinate transformations and instead exploits the special structure of the magnetic stream function. One key point here is that the equation for the magnetic stream function $\phi$ does not have any bad linear terms, as that of the equation for the magnetic field, which contains $-(\mathfrak{B}_2, 0)^t$. This approach allows us to uncover new stability mechanisms specific to the anisotropically dissipative MHD system.

We remark that the absence of horizontal dissipative mechanisms in system (1.3) precludes the derivation of stability estimates through standard energy methods. Motivated by the work of Deng, Wu, and Zhang [Reference Deng, Wu and Zhang10] on the 2D Boussinesq equations, we utilize the appropriate Fourier multiplier technique to capture the enhanced dissipation and inviscid damping associated with the structure $y\partial_{x}-\nu\partial_{yy}$. This approach yields a gain of $1/3$ of a horizontal derivative while simultaneously establishing a zero-order dissipation estimate represented via a Riesz transform-type operator for the nonzero Fourier modes.

The Fourier multiplier operator $\mathcal{M}(k,\xi)$ is defined as

(1.4)\begin{equation} \mathcal{M}=\mathcal{M}_{1}+\mathcal{M}_{2}+1, \end{equation}

where $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are given by

(1.5)\begin{equation} \begin{aligned} &\mathcal{M}_{1}(k,\xi)=\varphi\!\left(\varpi^{\frac{1}{3}}|k|^{-\frac{1}{3}}\operatorname{sgn}(k)\xi\right), \qquad &k\neq0,\\ &\mathcal{M}_{2}(k,\xi)=\arctan\!\left(\tfrac{\xi}{k}\right)+\tfrac{\pi}{2}, \qquad &k\neq0,\\ &\mathcal{M}_{1}(0,\xi)=\mathcal{M}_{2}(0,\xi)=0, \qquad &k=0, \end{aligned} \end{equation}

with $\varpi=\min\{\nu,\mu\}$. Here, $\varphi$ is a smooth, real-valued, non-decreasing function, $\varphi\in C^{\infty}(\mathbb{R})$, satisfying

\begin{equation*} 0\leqslant \varphi(x)\leqslant 1, \qquad 0\leqslant \varphi'(x)\leqslant \tfrac{1}{4}, \quad x\in\mathbb{R}, \qquad \varphi'(x)=\tfrac{1}{4}, \quad x\in[-1,1]. \end{equation*}

The operator $\mathcal{M}$ is thus a self-adjoint Fourier multiplier on $L^2(\Omega)$, and one readily checks that

\begin{equation*} 1 \leqslant \mathcal{M} \leqslant 2+\pi. \end{equation*}

From (1.4)–(1.5), we obtain the key coercivity estimate

(1.6)\begin{equation} \begin{aligned} k\partial_{\xi}\mathcal{M}+\nu\xi^{2}\;\geqslant\; (k\partial_{\xi}\mathcal{M}_{1}+\varpi\xi^{2})+k\partial_{\xi}\mathcal{M}_{2} \;\geqslant\;\tfrac{1}{4}\varpi^{\frac{1}{3}}|k|^{\frac{2}{3}}+\frac{k^{2}}{k^{2}+\xi^{2}}, \end{aligned} \end{equation}

valid for all $k\neq 0$, $\nu \gt 0$, and $\xi\in \mathbb{R}$. As a representative case, consider the vorticity equation for $\omega$. We deduce

(1.7)\begin{equation} \langle (y\partial_x-\nu \partial_{yy})\omega, \mathcal{M}\omega\rangle \;\geqslant\;\tfrac{1}{4}\,\varpi^{\frac{1}{3}}\|\,|D_x|^{\frac{1}{3}}\omega\|_{L^2}^2 +\|\,|D_{x}|(-\Delta)^{-\frac12}\omega^{\neq}\|_{L^2}^2. \end{equation}

Consequently, the following dissipation inequality holds for the linearized system:

\begin{equation*} \frac{d}{dt}\|\sqrt{\mathcal{M}}\omega\|_{L^{2}}^{2} +\nu\|\partial_{y}\omega\|_{L^{2}}^{2} +\tfrac{1}{4}\varpi^{\frac{1}{3}}\|\,|D_{x}|^{\frac{1}{3}}\omega\|_{L^{2}}^{2} +\|\,|D_{x}|(-\Delta)^{-\frac{1}{2}}\omega^{\neq}\|_{L^{2}}^{2}\;\leqslant\;0. \end{equation*}

To state our main result precisely, we first introduce several notations. For $t \geqslant 0$ and $b \in \mathbb{R}$, we define

\begin{equation*} \mathcal{M}^{b}_{t}:=\sqrt{\mathcal{M}}\,\Lambda^{b}_{t}, \qquad \mathcal{M}^{b}_{t}(k,\xi):=\sqrt{\mathcal{M}(k,\xi)}\,\Lambda^{b}_{t}(k,\xi), \end{equation*}

where the time-dependent elliptic operator $\Lambda^{b}_{t}$ is given for any $b\in \mathbb{R}$ and $t \geqslant 0$ by

\begin{equation*} \Lambda^{b}_{t}:=\big(1-\partial^{2}_{x}-(\partial_{y}+t\partial_{x})^{2}\big)^{\frac{b}{2}}, \end{equation*}

or equivalently, in Fourier space,

\begin{equation*} \Lambda^{b}_{t}(k,\xi):=\big(1+k^{2}+(\xi+tk)^{2}\big)^{\frac{b}{2}}. \end{equation*}

Here, the Fourier transform is defined by

\begin{equation*} \hat{f}(k,\xi):=\int_{\mathbb{R}}e^{-i\xi y}\int_{\mathbb{T}}e^{-ikx}f(x,y)\,dx\,dy. \end{equation*}

The operator $\Lambda^{b}_{t}$ was first introduced in [Reference Deng, Wu and Zhang10]. For each fixed $t \geqslant 0$, it behaves like the standard nonhomogeneous fractional differential operator $(I-\Delta)^{b/2}$. A key property of $\Lambda^{b}_{t}$ is its commutation with the transport operator $\partial_t + y \partial_x$, namely,

\begin{equation*} \Lambda^{b}_{t} (\partial_t + y \partial_x) = (\partial_t + y \partial_x)\Lambda^{b}_{t}. \end{equation*}

This commutation allows Sobolev norms to be estimated without breaking the intrinsic transport structure of the equations, and plays a crucial role in our analysis.

We also introduce notations for horizontal averages and oscillatory modes. For any function $f(x,y)$, we write

\begin{equation*} f=\bar{f}+f^{\neq}, \end{equation*}

where

\begin{equation*} \bar{f}:=\int_{\mathbb{T}} f(x,y)\,dx, \qquad f^{\neq}:=f-\bar{f}. \end{equation*}

The fractional derivative in the $x$-variable is defined in Fourier space by

\begin{equation*} \widehat{|D_x|^\gamma f}(k, \xi) = |k|^\gamma \hat f(k, \xi). \end{equation*}

Throughout this paper, $C$ denotes a positive constant independent of $\nu$, whose value may vary from line to line. We also adopt the shorthand $D=\tfrac{1}{i}\partial$ and

\begin{equation*} \|(f,g)\|_{L^{2}}:=\|f\|_{L^{2}}+\|g\|_{L^{2}}. \end{equation*}

These notations will be used throughout the paper to formulate our stability estimates and state the main theorems.

Our main result is stated as follows:

Theorem 1.1. Assume $b \gt 1$, $\varpi= \min\{\nu, \mu\}$, $\alpha\geqslant \frac{1}{2}$, $\beta\geqslant \alpha+\frac{2}{3}$, $\frac{5}{6}\leqslant\gamma\leqslant 2\alpha-\frac{1}{6}$ and $\nabla\cdot u_{0}=\nabla\cdot \mathfrak{B}_{0}=0$. Let $\omega_0$, $j_0$, and $\phi_0$ denote the initial vorticity, current density, and magnetic stream function, respectively. Then, there exists a sufficiently small positive constant $\varepsilon$, such that if the initial data $(\omega_{0},\phi_{0})$ satisfy

\begin{equation*} \|(\omega_{0},j_{0})\|_{H^{b}} \leqslant \varepsilon \varpi^{\alpha},\quad \||D_{x}|^{\frac{2}{3}}\phi_{0}\|_{H^{b}} \leqslant \varepsilon \varpi^{\beta},\quad \|\partial_{y}\bar{\phi}_{0}\|_{H^{b}} \leqslant \varepsilon \varpi^{\gamma}, \end{equation*}

then the Cauchy problem (1.3) admits a unique global solution $(\omega,\phi,j)$ satisfying

\begin{equation*} \begin{aligned} &\|\Lambda^{b}_{t}(\omega,j)\|_{L^{\infty}L^{2}} +\varpi^{\frac{1}{2}}\|\Lambda^{b}_{t}\partial_{y}(\omega,j)\|_{L^{2}L^{2}}+\varpi^{\frac{1}{6}}\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{1}{3}}(\omega,j)\|_{L^{2}L^{2}}\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\|\Lambda^{b}_{t}|D_{x}|(-\Delta)^{-\frac{1}{2}}(\omega,j)\|_{L^{2}L^{2}}\leqslant C\varepsilon \varpi^{\alpha},\\ &\|\Lambda^{b}_{t}|D_{x}|^{\frac{2}{3}}\phi\|_{L^{\infty}L^{2}} +\mu^{\frac{1}{2}}\|\Lambda^{b}_{t}\partial_{y}|D_{x}|^{\frac{2}{3}}\phi\|_{L^{2}L^{2}}+\varpi^{\frac{1}{6}}\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{1}{3}}|D_{x}|^{\frac{2}{3}}\phi\|_{L^{2}L^{2}}\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\|\Lambda^{b}_{t}|D_{x}|(-\Delta)^{-\frac{1}{2}}|D_{x}|^{\frac{2}{3}}\phi\|_{L^{2}L^{2}}\leqslant C\varepsilon \varpi^{\beta},\\ &\|\Lambda^{b}_{t}\partial_{y}\bar{\phi}\|_{L^{\infty}L^{2}} +\mu^{\frac{1}{2}}\|\Lambda^{b}_{t}\partial_{yy}\bar{\phi}\|_{L^{2}L^{2}}\leqslant C\varepsilon \varpi^{\gamma}, \end{aligned} \end{equation*}

where $C \gt 0$ is independent of $\nu,\mu,\alpha,\beta,\gamma$, and $\varepsilon$.

We add a remark on the restrictions imposed on $\alpha,\beta,\gamma$, and on their sharpness.

Remark 1.1. The restrictions on $\alpha,\beta,\gamma$ arise from the intrinsic anisotropic scaling of the linearized MHD–Couette system with vertical dissipation. The current equation (1.9) contains the linear stretching term $2\partial_{xy}\phi$, whose control through vertical diffusion produces an unavoidable $\mu^{-1}$ loss (see (1.11) and (2.38)). On the other hand, the transport–diffusion operator $y\partial_x-\nu\partial_{yy}$ yields a gain of $1/3$ horizontal derivative via enhanced dissipation (see (1.6)–(1.7)). These two linear effects determine the scaling of the nonlinear energy inequalities (2.6). Requiring that each nonlinear term carry a nonnegative power of $\varpi=\min\{\nu,\mu\}$ leads precisely to

\begin{equation*} \alpha\geqslant \tfrac12,\qquad \beta\geqslant \alpha+\tfrac23,\qquad \tfrac56\le \gamma\le 2\alpha-\tfrac16 . \end{equation*}

Hence the exponents in Theorem 1.1 are consistent with the anisotropic scaling of the problem and are essentially sharp within the present energy framework. Any substantial relaxation would require a mechanism compensating the $\mu^{-1}$ stretching loss or improving the $1/3$ enhanced dissipation gain.

We also remark that the explicit enhanced dissipation decay $e^{-c\nu^{1/3}t}$ can be incorporated into the estimates of Theorem 1.1.

Remark 1.2. The explicit enhanced dissipation rate $e^{-c\nu^{1/3}t}$ can be obtained within the framework of Theorem 1.1. In particular, for the nonzero Fourier modes we have

(1.8)\begin{equation} \|\Lambda^{b}_{t}(\omega,j)\|_{L^{2}} \le C\varepsilon \varpi^{\alpha}e^{-c\nu^{1/3}t},\qquad \|\Lambda^{b}_{t}|D_{x}|^{2/3}\phi\|_{L^{2}} \le C\varepsilon \varpi^{\beta}e^{-c\nu^{1/3}t}, \end{equation}

where $c \gt 0$ is a constant independent of $\nu,\mu,\alpha,\beta$, and $\varepsilon$. These bounds follow by incorporating the time weight $e^{-c\nu^{1/3}t}$ into the energy functionals. The enhanced dissipation rate $e^{-c\nu^{1/3}t}$ is already implicit in the linear coercivity estimate of the multiplier $\mathcal{M}$. More precisely, estimate (1.7) yields

\begin{equation*} \frac{d}{dt}\|\sqrt{\mathcal{M}}\,\omega\|_{L^2}^2 + c\nu^{1/3}\||D_x|^{1/3}\omega\|_{L^2}^2 \le 0, \end{equation*}

for some $c \gt 0$. Since $|k|\geqslant1$ on the nonzero Fourier modes, this implies

\begin{equation*} \frac{d}{dt}\|\omega_{\neq}\|_{L^2}^2 + c\nu^{1/3}\|\omega_{\neq}\|_{L^2}^2 \le 0, \end{equation*}

and hence

\begin{equation*} \|\omega_{\neq}(t)\|_{L^2} \le e^{-c\nu^{1/3}t}\|\omega_{\neq}(0)\|_{L^2}. \end{equation*}

A special consequence of Theorem 1.1 is the following corollary.

Corollary 1.2. Assume the same conditions as in Theorem 1.1. If the initial value $\phi_{0}$ satisfies

\begin{equation*} \|\phi_{0}\|_{H^{b}} \leqslant \varpi^{\lambda} \quad \text{for} \quad \lambda\geqslant 0, \end{equation*}

then the system admits a unique global solution $\phi$ satisfying

\begin{align*} \begin{aligned} &\|\Lambda^{b}_{t}\phi\|_{L^{\infty}L^{2}} +\mu\|\Lambda^{b}_{t}\partial_{y}\phi\|_{L^{2}L^{2}}+\varpi^{\frac{1}{3}}\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{1}{3}}\phi\|_{L^{2}L^{2}} \nonumber\\ & \quad +\|\Lambda^{b}_{t}|D_{x}|(-\Delta)^{-\frac{1}{2}}\phi\|_{L^{2}L^{2}}\leqslant C\varpi^{\lambda}. \end{aligned} \end{align*}

We now turn to the main analytical difficulties underlying Theorem 1.1. The foremost obstacle arises from the nonlinear term $\mathfrak{B}\cdot \nabla \Delta \phi$ in the vorticity equation of system (1.3). Controlling this term requires sharp estimates on higher-order derivatives of the magnetic stream function $\phi$. Since the current density is defined by $j:=\Delta \phi$, we derive the governing equation

(1.9)\begin{equation} \begin{aligned} \partial_{t}j+y\partial_{x}j -\mu\partial_{yy} j+2\partial_{xy}\phi=-u\cdot \nabla j+\mathfrak{B}\cdot \nabla \omega-Q(\nabla u,\nabla \mathfrak{B}), \end{aligned} \end{equation}

where the quadratic form is given by

(1.10)\begin{equation} \begin{aligned} Q(\nabla u,\nabla \mathfrak{B}):=2\partial_{x}u_{1}(\partial_{x}\mathfrak{B}_{2}+\partial_{y}\mathfrak{B}_{1})-2\partial_{x}\mathfrak{B}_{1}(\partial_{x}u_{2}+\partial_{y}u_{1}). \end{aligned} \end{equation}

The symmetry between vorticity $\omega$ and current density $j$ plays a crucial role: it enables us to balance the nonlinear advective terms $\mathfrak{B}\cdot\nabla \omega$ and $\mathfrak{B}\cdot\nabla j$, thereby recovering stability.

A further challenge stems from the additional linear term $2\partial_{xy}\phi$ appearing in (1.9), which complicates the energy estimates. To overcome this, we construct auxiliary energy functionals. Specifically,

(1.11)\begin{equation} \begin{aligned} \langle -\Lambda^{b}_{t}2\partial_{xy}\phi, \mathcal{M}\Lambda^{b}_{t}j\rangle\leqslant C\mu^{-1}\|\Lambda^{b}_{t}\partial_{x}\phi\|_{L^{2}}^{2}+\epsilon\mu \|\Lambda^{b}_{t}\partial_{y}j\|_{L^{2}}^{2}. \end{aligned} \end{equation}

The $\mu^{-1}$ factor in the leading term reveals the necessity of refined bounds on $\|\Lambda^{b}_{t}|D_{x}|\phi\|_{L^{2}L^{2}}$. To achieve this, we incorporate fractional horizontal derivatives of $\phi$ into the energy functional.

This refinement introduces yet another difficulty: nonlinear terms now require control of $\partial_{y}\bar{\phi}$. In fact,

(1.12)\begin{equation} \begin{aligned} &\langle \Lambda^{b}_{t}(\vert D_{x}\vert^{\frac{2}{3}}u\cdot \nabla \phi), \mathcal{M}\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}\phi\rangle\\ =&\langle \mathcal{M}^{b}_{t}(\vert D_{x}\vert^{\frac{2}{3}}u\cdot \nabla \phi^{\neq}), \mathcal{M}^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}\phi\rangle+\langle \mathcal{M}^{b}_{t}(\vert D_{x}\vert^{\frac{2}{3}}u\cdot \nabla \bar{\phi}), \mathcal{M}^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}\phi\rangle\\ \leqslant &C\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}\phi\|_{L^{2}}(\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}u_{2}\|_{L^{2}}\|\Lambda^{b}_{t}\partial_{y}\phi^{\neq}\|_{L^{2}}+\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}u_{1}\|_{L^{2}}\|\Lambda^{b}_{t}\vert D_{x}\vert\phi\|_{L^{2}})\\ &\ \ +C\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}\phi\|_{L^{2}}\|\Lambda^{b}_{t}\vert D_{x}\vert^{\frac{2}{3}}u_{2}\|_{L^{2}}\|\Lambda^{b}_{t}\partial_{y}\bar{\phi}\|_{L^{2}}. \end{aligned} \end{equation}

Thus, as indicated in (1.12), the control of nonlinear interactions necessitates an auxiliary functional tied to $\|\Lambda^{b}_{t}\partial_{y}\bar{\phi}\|_{L^{\infty}L^{2}}$.

We finally remark that one may alternatively analyse the vorticity–current system $(\omega,j)$ in sheared coordinates and treat the linear stretch term $2\partial_{xy}\phi$ using Fourier multipliers. We instead work with the magnetic stream function $\phi$, whose equation possesses a more favourable linear structure and commutes with the transport operator $\partial_t+y\partial_x$. Moreover, the nonlocal relation $\phi=\Delta^{-1}j$ is handled more directly in the present anisotropic (vertical-only) dissipation framework, particularly for the zero mode.

Recent years have seen substantial progress on stability and transition thresholds for the 2D MHD equations near Couette flow. In the presence of a constant background magnetic field, Dolce [Reference Dolce11] established the Sobolev stability threshold and enhanced dissipation for the 2D MHD equations using symmetric variables. Sobolev stability thresholds for partially dissipative MHD near Couette were obtained by Knobel and Zillinger [Reference Knobel and Zillinger22], who analysed regimes with anisotropic magnetic dissipation. Nonlinear stability and enhanced dissipation in Sobolev spaces for the 2D MHD system with uniform magnetic field were further proved by Jin, Ren, and Wei [Reference Knobel18]. Knobel and Zillinger [Reference Knobel and Zillinger21] analysed nonlinear resonance mechanisms for the 2D MHD equations with magnetic dissipation near a combination of Couette flow and constant background magnetic field. They constructed explicit global low-frequency solutions and showed that the linearized dynamics around these states exhibit high-frequency resonant interactions leading to echo chains, which produce Gevrey-class norm inflation and infinite-time blow-up in Sobolev regularity. More recently, Knobel [Reference Knobel20] developed a unified framework for Sobolev stability thresholds of 2D dissipative fluid equations, including the MHD equations near Couette flow, based on the suppression of nonlinear fluid-echo interactions. Complementary to these stability results, Dolce, Knobel, and Zillinger [Reference Dolce, Knobel and Zillinger12] investigated instability mechanisms for the viscous, non-resistive 2D MHD equations near Couette flow, proving large-norm inflation of the current despite viscosity. Their analysis also employs the magnetic stream-function formulation. All of the above works rely on sheared or moving coordinate transformations adapted to the Couette flow, whereas the present analysis avoids coordinate changes and instead exploits the special structure of the magnetic stream-function formulation.

The remainder of this paper is organized as follows. Section 2 establishes a key lemma for the a priori estimates, introduces several energy functionals, and applies a bootstrap argument to prove Theorem 1.1 through three propositions. Section 3 is devoted to the proof of Corollary 1.2.