The regularity of semi-hyperbolic patches near sonic curves for the two-dimensional compressible magnetohydrodynamic equations. (English) Zbl 07812938
Summary: 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 \(C^1\) continuous.
© 2020 Wiley-VCH GmbH
© 2020 Wiley-VCH GmbH
MSC:
| 35L65 | Hyperbolic conservation laws |
| 35J70 | Degenerate elliptic equations |
| 35R35 | Free boundary problems for PDEs |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
Keywords:
characteristic decompositions; 2D compressible MHD equations; regularity; self-similar solution; semi-hyperbolic patch; sonic curveReferences:
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