具有消失垂直磁阻的三维可压缩磁流体动力学方程在半空间中的全局一致正则性和大时间行为
Global uniform regularity and large time behavior of solutions to three dimensional compressible MHD equations with vanishing vertical magnetic resistivity in half space
AI总结：
研究具有消失垂直磁阻的三维可压缩MHD方程在半空间的解，利用各向异性Sobolev不等式等方法获得全局一致正则性估计，推导出解的衰减率，结合两者证明了消失垂直磁阻极限过程的显式时间一致\(L^2\)收敛速率为\(\varepsilon^{\frac{1}{4}}\)。
AI中文摘要：
本文旨在建立三维可压缩磁流体动力学（MHD）方程在半空间中具有消失垂直磁阻的解的全局正则性和大时间行为，在速度上具有无滑移边界条件，在磁场方面具有理想导电边界条件。通过利用各向异性Sobolev不等式和精细估计，我们能够获得与小垂直电阻率系数\(\varepsilon\)无关的解的全局一致正则性估计。这些估计使我们能够在\(\varepsilon \to 0\)时取极限，得到收敛到无垂直磁阻的相应MHD系统的全局时间解。此外，基于对相关线性化算子在半空间中的半群的详细分析和获得的一致能量估计，还推导出了原始系统解的\(H^{1}\left(\mathbb{R}_{+}^{3}\right)\)衰减率。与Gao和Xie \cite{JDE}得到的极限系统的衰减估计相比，目前的衰减估计显示出较慢的速率，这归因于边界层的出现导致缺乏高阶法向导数估计。最后，我们结合正则性和衰减结果，严格证明了对于消失垂直磁阻极限过程的显式时间一致\(L^2\)收敛速率为\(\varepsilon^{\frac{1}{4}}\)。
英文摘要：
This paper aims to establish the global regularity and large time behavior of solutions to the three-dimensional (3D) compressible magnetohydrodynamics (MHD) equations with vanishing vertical magnetic resistivity in the upper half-space with no-slip boundary condition on velocity and perfectly conducting boundary condition on magnetic field. By exploiting anisotropic Sobolev inequalities and elaborated estimates, we are able to achieve global-in-time uniform regularity estimates of solutions, which are independent of small vertical resistivity coefficient $\varepsilon$. These uniform global regularity estimates allow us to pass to the limit as \(\varepsilon \to 0\) and obtain the convergence to the corresponding MHD system without vertical magnetic resistivity globally in time. Moreover, the $H^{1}\left(\mathbb{R}_{+}^{3}\right)$ decay rates of solutions to the original system are also derived based on the detailed analysis on semigroup of the related linearized operator in half space and the uniform energy estimates achieved. In contrast to the decay estimates for the limit system obtained by Gao and Xie \cite{JDE}, the present decay estimate exhibits a slower rate, attributable to the absence of higher-order normal derivative estimates, which results from the occurrence of boundary layers. Finally, we combine both sets of regularity and decay results to rigorously prove an explicit time-uniform $L^2$ convergence rate of order $\varepsilon^{\frac{1}{4}}$ for the vanishing vertical magnetic resistivity limit process.
