On the Global Existence and Stability of a Multi-Dimensional Supersonic Conic Shock Wave

Abstract

We establish the global existence and stability of a three-dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle. The flow is described by a 3-D steady potential equation, which is multi-dimensional, quasilinear, and hyperbolic with respect to the supersonic direction. Making use of the geometric properties of the pointed shock surface together with the Rankine–Hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone, we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new Hardy-type inequality on the shock surface. Based on this, we prove that a multi-dimensional conic shock attached to the vertex of the cone exists globally when the Mach number of the incoming supersonic flow is sufficiently large. Moreover, the asymptotic behavior of the 3-D supersonic conic shock solution, which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone, is also explicitly given.