发表机构
School of Mathematics and Statistics, Shandong University of Technology; Department of Mathematics, University of Notre Dame（山东理工大学数学与统计学院; 圣母大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对仅含流体粘性的三维全可压缩非电阻无热传导MHD系统，证明在小扰动、背景磁场满足丢番图条件时存在全局光滑解及代数衰减率，揭示了隐藏耗散机制并处理了大密度振荡问题。
AI 中文摘要
我们研究周期环面$\u2119^3$上的三维全可压缩磁流体动力学（MHD）系统，其仅有的耗散机制是流体的粘性：磁场为非电阻性，流动不具备热传导性。我们证明，当平衡态$(\u2009\boldsymbol{0},\bar P,\boldsymbol{n})$的扰动$(\boldsymbol{u}_0,\u2009P_0-\bar P,\u2009\boldsymbol{H}_0-\boldsymbol{n})$在高阶Sobolev空间中足够小，且背景磁场$\boldsymbol{n}\u2208\u211d^3$满足丢番图条件时，该系统存在唯一的全局光滑解，并具有显式代数衰减速率。初始密度无需满足任何小性条件：仅要求其远离真空且有上界，可具有任意大的振荡。证明揭示了一种隐藏的耗散机制。尽管密度、压强和磁场本身都不具备任何扩散或阻尼，但这些量通过背景场$\boldsymbol{n}$与速度的耦合，结合源于丢番图条件的庞加莱型不等式，可为压强和磁场扰动生成有效耗散。密度的大振荡通过两层能量论证处理：其中中间阶能量的加权时间衰减估计恰好补偿密度最高阶范数的线性时间增长。
英文摘要
We consider the three-dimensional full compressible magnetohydrodynamic(MHD) system on the periodic torus $\mathbb T^3$ in the regime where the only dissipative mechanism acting on the system is the viscosity of the fluid: the magnetic field is non-resistive and the flow is non-heat-conducting. We prove that this system admits a unique global smooth solution, together with explicit algebraic decay rates, provided that the perturbation $(\mathbf u_0,\,P_0-\bar P,\,\mathbf H_0-\mathbf n)$ of the equilibrium state $(\mathbf 0,\bar P,\mathbf n)$ is sufficiently small in a high-order Sobolev space and the background magnetic field $\mathbf n\in\mathbb R^3$ satisfies a Diophantine condition. No smallness whatsoever is imposed on the initial density: it is only required to be bounded away from vacuum and from infinity, and may exhibit arbitrarily large variations. The proof uncovers a hidden dissipation mechanism. Although neither the density, nor the pressure, nor the magnetic field is endowed with any diffusion or damping of its own, the coupling of these quantities with the velocity through the background field $\mathbf n$, combined with a Poincaré-type inequality of Diophantine origin, generates effective dissipation for both the pressure and the magnetic field perturbations. The large variations of the density are handled by a two-tier energy argument, in which weighted time-decay estimates for the intermediate-order energy compensate exactly for the linear-in-time growth of the highest-order norm of the density.
