On the existence and regularity of solutions of semi-hyperbolic patches to 2-D Euler equations with van der Waals gas

TL;DR

This paper proves the existence and regularity of semi-hyperbolic patch solutions for 2-D Euler equations with van der Waals gas, relevant to transonic flows and numerical Riemann problem solutions.

Contribution

It establishes the global smoothness and regularity of solutions and sonic curves, using characteristic decomposition and bootstrap methods, for the first time in this context.

Findings

Existence of global smooth solutions with $C^{1, 1/2}$ regularity.

Sonic curve is $C^{1, 1/2}$ continuous.

Shock formation occurs as an envelope for positive characteristics.

Abstract

This article is concerned in establishing the existence and regularity of solution of semi-hyperbolic patch problem for two-dimensional isentropic Euler equations with van der Waals gas. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is very common in the numerical solution of Riemann problems. We use the idea of characteristic decomposition and bootstrap method to prove the existence of global smooth solution which is uniformly $C^{1, \frac{1}{2}}$ continuous up to the sonic curve. We also prove that the sonic curve is $C^{1, \frac{1}{2}}$ continuous. Further, we show the formation of shock as an envelope for positive characteristics before reaching their sonic points.