# Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones

## TL;DR

This paper investigates inverse problems in supersonic flows around cones, focusing on the stability of shocks and the existence of global solutions.

## Contribution

The paper introduces a modified Glimm-type scheme and a functional to analyze the stability of oblique conical shocks in supersonic flows.

## Key findings

- A modified Glimm-type scheme is developed to construct approximate solutions for inverse problems in supersonic flows.
- The existence of global entropy solutions with bounded BV norm is established under certain flow conditions.
- Entropy solutions asymptotically approach self-similar solutions determined by incoming flow and pressure.

## Abstract

We are concerned with inverse problems for supersonic potential flows past infinite axisymmetric Lipschitz cones. The supersonic flows under consideration are governed by the steady isentropic Euler equations for axisymmetric potential flows, which give rise to a singular geometric source term. We first study the inverse problem for the stability of an oblique conical shock as an initial-boundary value problem with both the generating curve of the cone surface and the leading conical shock front as free boundaries. We then establish the existence and asymptotic behavior of global entropy solutions with bounded BV norm of the inverse problem, under the condition that the Mach number of the incoming flow is sufficiently large and the total variation of the pressure distribution on the cone is sufficiently small. To this end, we first develop a modified Glimm-type scheme to construct approximate solutions by self-similar solutions as building blocks to balance the influence of the geometric source term. Then we define a Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions. Meanwhile, the approximate generating curves of the cone surface are also constructed. Next, when the Mach number of the incoming flow is sufficiently large, by asymptotic analysis of the reflection coefficients in those interaction estimates, we prove that appropriate weights can be chosen so that the corresponding Glimm-type functional decreases in the flow direction. Finally, we determine the generating curves of the cone surface and establish the existence of global entropy solutions containing a strong leading conical shock, besides weak waves. Moreover, the entropy solution is proved to approach asymptotically the self-similar solution determined by the incoming flow and the asymptotic pressure on the cone surface at infinity.