Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

Jianjun Chen; Geng Lai

doi:10.3934/cpaa.2019046

Article Contents

2019, Volume 18, Issue 2: 943-958. Doi: 10.3934/cpaa.2019046

This issue Previous Article Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation Next Article Fractal analysis of canard cycles with two breaking parameters and applications

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

Jianjun Chen^1, and
Geng Lai^2, ,

1.
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023, China
2.
Department of Mathematics, Shanghai University, Shanghai, 200444, China

* Corresponding author
* Corresponding author

Received: March 31, 2018

Revised: April 30, 2018

Published: October 2018

Supported by NSF of China (11301326) and the grant of the first-class Discipline of Universities in Shanghai

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Abstract

We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of flow patches appear frequently in transonic flow problems. In order to use the method of characteristic decomposition to construct such a flow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.

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Mathematics Subject Classification: Primary: 35L65, 35J70, 35R35; Secondary: 35J65.

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Figure 1. Pseudosonic curve. (Ⅰ) Keldysh type; (Ⅱ) Tricomi type

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Figure 2. A semi-hyperbolic patch

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Figure 3. Left: small Goursat problem; right: global solution

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