On Inverse Problems for Two-Dimensional Steady Supersonic Euler Flows past Curved Wedges

TL;DR

This paper studies the inverse problem of determining wedge boundaries in 2D steady supersonic Euler flows, establishing existence, stability, and uniqueness of solutions under small perturbations using wave-front tracking and Lyapunov functionals.

Contribution

It introduces a novel wave-front tracking approach and a Lipschitz continuous Lyapunov functional to prove existence and stability of wedge boundaries in inverse supersonic flow problems.

Findings

Existence of wedge boundaries for pressure conditions within a critical range.

Lipschitz stability of wedge boundaries with respect to flow and pressure perturbations.

Quantitative estimates of boundary and solution differences based on initial data perturbations.

Abstract

We are concerned with the well-posedness of an inverse problem for determining the wedge boundary and associated two-dimensional steady supersonic Euler flow past the wedge, provided that the pressure distribution on the boundary surface of the wedge and the incoming state of the flow are given. We first establish the existence of wedge boundaries and associated entropy solutions of the inverse problem when the pressure on the wedge boundary is larger than that of the incoming flow but less than a critical value, and the total variation of the incoming flow and the pressure distribution is sufficiently small. This is achieved by carefully constructing suitable approximate solutions and approximate boundaries via developing a wave-front tracking algorithm and the rigorous proof of their strong convergence to a global entropy solution and a wedge boundary respectively. Then we establish the $L^{\infty}$--stability of the wedge boundaries, by introducing a modified Lyapunov functional for two different solutions with two distinct boundaries, each of which may contain a strong shock-front. The modified Lyapunov functional is carefully designed to control the distance between the two boundaries and is proved to be Lipschitz continuous with respect to the differences of the incoming flow and the pressure on the wedge, which leads to the existence of the Lipschitz semigroup. Finally, when the pressure distribution on the wedge boundary is sufficiently close to that of the incoming flow, using this semigroup, we compare two solutions of the inverse problem in the respective supersonic full Euler flow and potential flow and prove that, at $x>0$, the distance between the two boundaries and the difference of the two solutions are of the same order of $x$ multiplied by the cube of the perturbations of the initial boundary data in $L^\infty\cap BV$.