Supersonic flow of Chaplygin gas past a delta wing

TL;DR

This paper analyzes the complex supersonic flow of a Chaplygin gas around a delta wing, addressing the mathematical challenges of mixed-type equations and establishing the existence and uniqueness of solutions.

Contribution

It introduces a novel approach to handle nonlinear mixed-type equations in three-dimensional supersonic flows, proving the unique existence of solutions for the problem.

Findings

Established a Lipschitz estimate for the nonlinear equation

Proved the unique existence of the flow solution

Addressed the mathematical complexity of mixed-type equations

Abstract

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or rarefaction wave attached to the leading edges. The flow under study is described by the three-dimensional steady Euler system. In conical coordinates, this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type. The type of this equation depends fully on the solutions of the problem itself, and thus it cannot be determined in advance. We overcome the difficulty by establishing a crucial Lipschitz estimate, and finally prove the unique existence of the solution via the method of continuity.