Three-dimensional stationary supersonic flows with axial symmetry in relativistic hydrodynamics

The study of supersonic flows in relativistic hydrodynamics (RHDs) is of some interest for cosmology. This paper studies two types of three-dimensional (3D) stationary supersonic flows with axial symmetry in RHDs: supersonic flows expanding in vacuum and supersonic flows against an infinite curved cone. To study supersonic flows expanding in vacuum, we consider a Cauchy problem for the 3D steady RHD equations with initial data that are a combination of an axisymmetric supersonic flow in a circular domain and vacuum in the remaining domain. Global existence of a piecewise smooth solution expanding in vacuum to the Cauchy problem is obtained, provided that the initial mass-energy density is sufficiently small. When a supersonic flow arrives and hits a curved cone, a shock wave will arise around the cone. Moreover, if the vertex angle of the cone is less than a critical angle, then the shock will be attached to the vertex of the cone. We obtain the global existence of attached conical shock waves, provided that the curved cone satisfies some suitable conditions and the speed of the incoming flow is close to a limit speed.