发表机构
Beijing Normal University; China University of Mining and Technology; University of Notre Dame（北京师范大学; 中国矿业大学; 圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了三维可压缩非电阻MHD系统在无Diophantine条件下小对称扰动的全局稳定性，通过揭示总压力的隐藏阻尼波结构克服无耗散障碍，获得一致稳定与多项式衰减。
AI 中文摘要
我们证明了在环面 $\mathbb{T}^3$ 上，三维可压缩粘性非电阻磁流体动力学系统在平衡态 $(1,\mathbf{0},\mathbf{e}_3)$ 附近的经典解的全局适定性和非线性稳定性，适用于小的 $x_3$-对称扰动，且对背景磁场没有任何 Diophantine 条件。核心障碍在于 $x_3$-独立扇区，在该扇区中密度和磁场没有耗散、没有阻尼、也没有衰减。我们通过揭示精确总压力 $\mathcal{D}=P(1+a)+B_3+\frac12|\mathbf{B}|^2$ 的隐藏波结构来克服这一障碍：平均对 $(\overline{\mathbf{u}},\overline{\mathcal{D}})$ 满足一个封闭系统，且 $\overline{\mathcal{D}}$ 满足一个强阻尼波动方程，其主波部分以快速磁声速传播，而其非抛物分支以不涉及导数增益的速率衰减。这种隐藏阻尼替代了缺失的磁场耗散，同时吸收了由可压缩性产生的主要障碍——磁压力 $\frac12\nabla|\mathbf{B}|^2$。结合振荡扇区的阻尼波结构以及具有平移时间权重的时空加权能量泛函，这得到了全局存在性、一致稳定性和显式多项式衰减率。
英文摘要
We prove the global well-posedness and nonlinear stability of classical solutions to the three-dimensional compressible viscous, non-resistive MHD system on $\mathbb{T}^3$ near the equilibrium $(1,\mathbf{0},\mathbf{e}_3)$, for small $x_3$-symmetric perturbations and without any Diophantine condition on the background magnetic field. The central obstruction is the $x_3$-independent sector, in which the density and the magnetic field possess no dissipation, no damping, and no decay. We overcome it by exhibiting a hidden wave structure for the exact total pressure $\mathcal{D}=P(1+a)+B_3+\frac12|\mathbf{B}|^2$, the averaged pair $(\overline{\mathbf{u}},\overline{\mathcal{D}})$ obeys a closed system, and $\overline{\mathcal{D}}$ satisfies a strongly damped wave equation whose principal wave part propagates at the fast magnetosonic speed and whose non-parabolic branch damps at a rate that involves no gain of derivatives. This hidden damping substitutes for the missing magnetic dissipation, and simultaneously absorbs the magnetic pressure $\frac12\nabla|\mathbf{B}|^2$, the main obstruction created by compressibility. Together with a damped wave structure for the oscillatory sector and space-time weighted energy functionals with shifted time weights, this yields global existence, uniform stability, and explicit polynomial decay rates.
