关于不可压缩磁流体动力学的粘性消失和磁扩散极限
On the Vanishing Viscosity and Magnetic Diffusion Limit for Incompressible Magnetohydrodynamics
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中文总结 AI 辅助
研究不可压缩磁流体动力学方程的无粘和无电阻极限，通过推导充分条件、建立准则及分析衰减率，得到MHD系统解的极限及磁普朗特数极限，是该问题数学上的首个严格结果。
中文摘要 AI 辅助
本文聚焦于有界域中不可压缩磁流体动力学（MHD）方程的无粘和无电阻极限。在内部正则性一致和边界等度连续性假设下，推导了MHD系统Leray - Hopf解的极限的充分条件，当\(\mu,\nu\rightarrow0\)时收敛到理想MHD系统的弱解。建立了类似Kato型准则表明能量耗散率与粘性和电阻率系数无关。还分析了粘性和电阻率的不同衰减率，得到了不可压缩MHD系统无限和消失磁普朗特数（\(Pm=\frac{\mu}{\nu}\)）的极限。据作者所知，这是关于磁普朗特数极限的首个数学上的严格结果。
英文摘要
The article focuses on the inviscid and non-resistive limit of the incompressible magnetohydrodynamics (MHD) equations in a bounded domain. We derive sufficient conditions for the limit of Leray-Hopf solutions of the MHD system, which converges to the weak solutions of the ideal MHD system when $μ,ν\rightarrow0$, under the assumptions of uniform interior regularity and uniform equi-continuity at the boundary. Therefore, we establish an analogue of the Kato-type criterion to show that the energy dissipation rate is independent of the coefficients of viscosity and resistivity. Moreover, to analyze the different decay rates for the viscosity and resistivity, we obtain the limits of infinite and vanishing magnetic Prandtl number ($Pm=\fracμν$) for the incompressible MHD system. To the best of the authors' knowledge, this is the first rigorous result from a mathematical viewpoint on the limit of the magnetic Prandtl number.
