Supersonic sonic patch solution for the two-dimensional Euler equations with a van der Waals equation of state

TL;DR

This paper constructs and analyzes a supersonic sonic patch solution for the two-dimensional Euler equations with a van der Waals equation of state, extending the theory from ideal gases to non-ideal, realistic gas models.

Contribution

It introduces a novel analytical framework for supersonic transonic flows with non-ideal gas effects, including existence and regularity results near sonic boundaries.

Findings

Existence of a globally defined supersonic solution

Uniform regularity up to the sonic boundary

Extension of sonic patch theory to non-ideal gases

Abstract

We investigate supersonic transonic phenomena in the two-dimensional compressible Euler equations governed by a polytropic van der Waals equation of state. In contrast to the ideal gas setting, the non-ideal pressure law introduces stronger nonlinear effects and modifies the degeneracy structure near sonic states, which significantly complicates the analytical treatment of transonic flows. Within the self-similar framework associated with the four-state Riemann problem, we construct a supersonic sonic patch solution that connects a strictly supersonic region to a sonic boundary along a pseudo streamline. The analysis is based on a characteristic decomposition combined with a partial hodograph transformation, through which the problem is reformulated as a degenerate hyperbolic system. We establish the existence of a globally defined supersonic solution and prove its uniform regularity up to the sonic curve. In addition, we investigate the regularity properties of the resulting sonic boundary. Our results extend the theory of supersonic sonic patches from polytropic gases to a realistic non-ideal gas model.