Regularity of solutions to the Navier-Stokes equations for compressible barotropic flows on a polygon

Abstract

The Navier-Stokes system for a steady-state barotropic nonlinear compressible viscous flow, with an inflow boundary condition, is studied on a polygon D. A unique existence for the solution of the system is established. It is shown that the lowest order corner singularity of the nonlinear system is the same as that of the Laplacian in suitable L-q spaces. Let w be the interior angle of a vertex P of D. If alpha := pi/w < 2 and q > 2/2-alpha then the velocity u is split into singular and regular parts near the vertex P. If alpha < 2 and 2 < q < 2/2-alpha or if alpha > 2 and 2 < q < < infinity, it is shown that u is an element of (H-2,H-q (D))(2).