This paper is concerned with the regularity of semi‐hyperbolic patches near sonic curves for the two‐dimensional (2D) compressible magnetohydrodynamic (MHD) equations. A semi‐hyperbolic patch is a flow in a region in which one family out of two families of wave characteristics start on sonic curve and end on transonic shock. This type of flow patterns appear frequently in solutions of 2D Riemann problems and transonic flow problems. In a recent study by Chen and Lai (Commun. Pure Appl. Anal. 18, 943–958 (2019)), we constructed a semi‐ hyperbolic patch for the 2D compressible MHD equations. In this paper, we derive a group of characteristic decompositions for the 2D MHD equations and show that the solution constructed in Chen and Lai (Commun. Pure Appl. Anal. 18, 943–958 (2019)) is smooth up to the sonic curve and the sonic curve is C¹ continuous.

中文翻译：

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二维可压缩磁流体动力学方程在声波曲线附近的半双曲斑的规则性

本文关注二维（2D）可压缩磁流体动力学（MHD）方程在声波曲线附近的半双曲补丁的规律性。半双曲修补程序是一个区域中的一个流，其中两个波特性家族中的一个以声波曲线开始，以跨音速冲击结束。这种类型的流动模式经常出现在2D Riemann问题和跨音速流动问题的解决方案中。在Chen和Lai的最新研究中（Commun。Pure Appl.Anal.18，943–958（2019）），我们为2D可压缩MHD方程构造了一个半双曲补丁。在本文中，我们导出了二维MHD方程的一组特征分解，并表明Chen和Lai（Commun.Pure Appl.Anal.18，943-958（2019））中构造的解决方案在声波曲线上都是平滑的声波曲线是C ¹ 连续。