Stability of Transonic Shocks in Steady Supersonic Flow past Multidimensional Wedges

TL;DR

This paper establishes the stability of transonic shocks in steady supersonic flow past multidimensional wedges using a nonlinear approach, free boundary formulation, and fixed boundary value problem analysis.

Contribution

It develops a nonlinear method to prove the stability of M-D transonic shocks, extending the analysis beyond the 2-D case with a novel fixed boundary problem approach.

Findings

Proved stability of weak shock solutions with transonic shocks in M-D flow.

Formulated the stability as a nonlinear elliptic free boundary problem.

Developed a contractive nonlinear iteration scheme for the problem.

Abstract

We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.