Interior jump and contact singularity for compressible flows with inflow jump datum

Abstract

Initial jump datum at a point must generate a jump curve directing into the region by the hyperbolic character of the continuity equation and the pressure gradient (in the momentum equation) is not well-defined. For this we construct a velocity vector field directed by the pressure difference on the jump curve. In bounded domains the jump curve must meet some points of the boundary and brings us contact singularities. In a rectangle domain we split the velocity vector field into a non-smooth part plus a contact singular one plus a smoother one and similar for the density function. We show piecewise regularity for the Navier-Stokes equations of compressible viscous flows and derive the Rankine-Hugoniot conditions.