Interior Jump and Regularity of Compressible Viscous Navier-Stokes Flows through a cut

Abstract

We study the stationary compressible Navier-Stokes equations in a domain containing an interior cut. The cut is an internal boundary having the vertices with a 2 pi angle at the tip points. It is a non-Lipschitz boundary. We prove existence and piecewise regularity by splitting the corner singularity functions at the cut tips and show that the continuity equation is solved by the density function having an interior jump discontinuity across the curve emanating from the rear tip. The piecewise regularity results from a vector function corresponding to the gradient of the jump density. The integral curve through the cut has the curvature that blows up at the cut tips, with the singularity order -1/2, which is the inverse to the corner singularity order at the cut tips.