发表机构
School of Mathematics, South China University of Technology; Department of Mathematics, University of Notre Dame; School of Mathematics and Statistics, Guangdong University of Technology（华南理工大学数学学院; 圣母大学数学系; 广东工业大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对垂直方向仅存在粘性和磁扩散的二维不可压磁流体库埃特流，研究其线性增长特性，引入适配剪切的傅里叶乘子分析非线性稳定性，在合适条件下证明全局非线性稳定性并给出扰动的衰减速率。
AI 中文摘要
我们研究了二维不可压磁流体系统在库埃特平衡态$\bigl((y,0)^{\rm T},(\beta,0)^{\rm T}\bigr)$附近的情况，该系统定义在$\mathbb T\times\mathbb R$上，处于粘性和磁扩散率仅作用于垂直方向的各向异性区域。对于带有$|\beta|>1/2$的无粘线性化问题，我们证明了涡量和电流密度随时间呈线性增长的尖锐结果，增长速率为时间的一次方。相比之下，速度和磁扰动的水平分量保持一致有界，而垂直分量则以$\langle t\rangle^{-1}$的速率表现出定量的无粘阻尼。对于非线性问题，我们引入了适配剪切的傅里叶乘子，该乘子可同时捕捉增强阻尼、临界时间效应以及回波型共振相互作用。在合适的水平背景磁场以及磁场强度、粘性和磁扩散率之间满足定量相容性条件的情况下，我们为满足$ \left\| \bigl( \mathbf v_{\rm in}-(y,0)^{\rm T}, \mathbf H_{\rm in}-(\beta,0)^{\rm T} \bigr) \right\|_{H^N} \leq \varepsilon_0\min\{\mu,\nu\}^{1/2}, N\geq4$的无散扰动建立了全局非线性稳定性。此外，非零水平傅里叶模式以增强阻尼速率$e^{-c\min\{\mu,\nu\}^{1/3}\,t},$在适配剪切的$H^N$范数中衰减，且垂直速度和磁分量获得额外的无粘阻尼因子$\langle t\rangle^{-1}$。
英文摘要
We study the two-dimensional incompressible magnetohydrodynamic system near the Couette equilibrium $\bigl((y,0)^{\rm T},(β,0)^{\rm T}\bigr)$ on $\mathbb T\times\mathbb R$, in the anisotropic regime where both viscosity and magnetic diffusivity act only in the vertical direction. For the inviscid linearized problem with $|β|>1/2$, we prove sharp linear-in-time growth of the vorticity and current density at the level of the time rate. In contrast, the horizontal components of the velocity and magnetic perturbations remain uniformly bounded, while the vertical components exhibit quantitative inviscid damping at the rate $\langle t\rangle^{-1}$. For the nonlinear problem, we introduce shear-adapted Fourier multipliers that simultaneously capture enhanced dissipation, critical-time effects, and echo-type resonant interactions. Under a suitable horizontal background magnetic field and a quantitative compatibility condition between the magnetic field strength, viscosity, and magnetic diffusivity, we establish global nonlinear stability for divergence-free perturbations satisfying $ \left\| \bigl( \mathbf v_{\rm in}-(y,0)^{\rm T}, \mathbf H_{\rm in}-(β,0)^{\rm T} \bigr) \right\|_{H^N} \leq \varepsilon_0\min\{μ,ν\}^{1/2}, N\geq4$. Moreover, the nonzero horizontal Fourier modes decay in a shear-adapted $H^N$ norm at the enhanced-dissipation rate $e^{-c\min\{μ,ν\}^{1/3}\,t},$ and the vertical velocity and magnetic components gain an additional inviscid-damping factor $\langle t\rangle^{-1}$.
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