Regularities of the Navier-Stokes equations for incompressible or compressible viscous flows on bounded singular domains

Abstract

In this dissertation we study incompressible or compressible viscous Navier-Stokes equations. For incompressible Navier-Stokes equations we use a planedomain with non-convex corners and assign the non-standard boundary conditions on its boundaries. We construct the corner singularity functions for the Stokes operator with zero vorticity and velocity normal componentboundary conditions, subtract the corner singularities from the solution and show increased regularity for the remainder. For compressible viscous Navier-Stokes equations we use a finite non-convex polyhedral cylinder in R^3 and assign the Dirichlet boundary conditions on its boundaries. We split the edge singularity from the velocity solution and show the H^{2,q}xH^{1,q}-regularityfor the velocity remainder and the pressure where 3 < q < 1/(1-π/β) and β is the angle of the edge. The edge flux coefficient is well-defined in H^{2/q'-π/β,q}(-1,1) and the pressure singularity is propagated along the streamline and its derivatives blow up across the streamlines.