The regularity of semi-hyperbolic patches of solutions to the two-dimensional compressible Euler equations in magnetohydrodynamics

The semi-hyperbolic patches appear frequently in solutions of multi-dimensional Riemann problem and transonic flow problems. We have obtained a semi-hyperbolic patch solution for the two-dimensional compressible magnetohydrodynamic equations (Chen and Geng, 2019 [4]). Subsequently, we prove the semi-hyperbolic patch solution is smooth up to the sonic curve and sonic curve is C-1 continuous (Chen and Geng, 2020 [5]). This paper will further consider the regularity of semi-hyperbolic patch problem to the two-dimensional compressible Euler equations in magnetohydrodynamics. By constructing an appropriate variable and using characteristic decomposition and bootstrap method, we show that the semi-hyperbolic patch solution is C-1 ,C- 1/6 up to the sonic curve and sonic curve is also C-1 ,C- 1/6. (c) 2025 Published by Elsevier Inc.