Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls

TL;DR

This paper proves the existence of global entropy solutions for steady supersonic reacting Euler flows past Lipschitz bending walls, using wave-front tracking and Glimm-type functionals, accounting for heat release and boundary perturbations.

Contribution

It introduces a modified wave-front tracking scheme and a Glimm-type functional to handle chemical reactions and boundary perturbations in supersonic flow models.

Findings

Existence of global entropy solutions under small total variation conditions.

Construction of approximate solutions with uniform total variation bounds.

Analysis of the asymptotic behavior of solutions in the flow direction.

Abstract

We are concerned with the two-dimensional steady supersonic reacting Euler flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the flow direction. We employ the modified wave-front tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that contains a strong rarefaction wave generated by the Lipschitz bending wall. In addition, the asymptotic behavior of the entropy solution in the flow direction is also analyzed.