Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows past Lipschitz Perturbed Cones

TL;DR

This paper proves the stability and existence of conical shocks in three-dimensional steady supersonic isothermal flows past Lipschitz perturbed cones, using a modified Glimm scheme and entropy solutions with BV regularity.

Contribution

It develops a novel Glimm-type scheme and establishes the global stability and asymptotic behavior of conical shocks in complex geometric flow settings.

Findings

Existence of global entropy solutions with conical shocks for large Mach numbers.

Asymptotic convergence of solutions to self-similar flow patterns.

Stability results depend on the smallness of cone slope variations.

Abstract

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.