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

Anamika Pandey, T. Raja Sekhar

TL;DR

<3-5 sentence high-level summary> The paper develops a rigorous framework for a supersonic–sonic patch in the two-dimensional compressible Euler system with a non-ideal van der Waals equation of state, addressing the degeneracy near sonic states. It combines a characteristic decomposition in angle variables with a partial hodograph transformation to reformulate the problem as a degenerate hyperbolic system, proving global existence and uniform regularity up to the sonic curve. The authors show that the sonic boundary is $C^{1,\mu}$ for any $\mu\in(0,1/3)$ and that the self-similar solution remains uniformly regular in a neighborhood of the sonic transition, thus extending the transonic patch theory from polytropic to non-ideal gases. This work provides a solid mathematical foundation for transonic structures in realistic gas models and suggests avenues for applying the approach to other equations of state and multidimensional flow configurations.

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.

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

TL;DR

<3-5 sentence high-level summary> The paper develops a rigorous framework for a supersonic–sonic patch in the two-dimensional compressible Euler system with a non-ideal van der Waals equation of state, addressing the degeneracy near sonic states. It combines a characteristic decomposition in angle variables with a partial hodograph transformation to reformulate the problem as a degenerate hyperbolic system, proving global existence and uniform regularity up to the sonic curve. The authors show that the sonic boundary is for any and that the self-similar solution remains uniformly regular in a neighborhood of the sonic transition, thus extending the transonic patch theory from polytropic to non-ideal gases. This work provides a solid mathematical foundation for transonic structures in realistic gas models and suggests avenues for applying the approach to other equations of state and multidimensional flow configurations.

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.
Paper Structure (18 sections, 18 theorems, 169 equations, 2 figures)

This paper contains 18 sections, 18 theorems, 169 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary analysis and problem formation
  3. Euler system in self-similar coordinates
  4. Problem formulation and statement of the main theorem
  5. Characteristic decompositions in terms of angle variables
  6. Sonic-supersonic boundary data and the main conclusion in terms of $(\theta, \bar{\omega})$
  7. The boundary data for $({X}, {Y})$
  8. Solution in a partial hodograph plane
  9. Reformulation in a partial hodograph plane
  10. The degenerate system and uniform bounds
  11. Construction of a determinate domain and the $C^0$-estimate
  12. Existence of global solution in $\Omega$
  13. Regularity of the solution
  14. Regularity of solution in the self-similar plane
  15. Inversion
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\widehat{LM}$ be a smooth pseudo streamline given by $\eta=\psi(\xi)$ for $\xi\in[\xi_1,\xi_2]$, where $\psi$ is strictly decreasing and concave. Assume that the pseudo Mach number $M_a$ is strictly decreasing along $\widehat{LM}$ and satisfies $M_a=1$ at the endpoint $M=(\xi_2,\psi(\xi_2))$. S

Figures (2)

  • Figure 1: A supersonic-sonic patch in the self-similar plane
  • Figure 2: The region $M'N'O'$

Theorems & Definitions (27)

  • Theorem 2.1
  • Lemma 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Corollary 3.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 17 more