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

Abstract

The steady-state nonlinear compressible viscous Navier-Stokes system with nonzero boundary conditions is considered on a polygon D. It is shown that the leading corner singularities for the velocity are the same as those of the Lame system and the leading corner singularity for the temperature is the same as that of the Laplacian. If P is a concave vertex of D with interior angle., the velocity u and temperature sigma can be split into singular and regular parts near the vertex P. The regular functions are u(R) = u - chi[C1r(gimel1)T(1)(theta) + C(2)r(gimel2)T(2)(theta)] is an element ofH(2,q) and sigma(R) = sigma - C-chi(3r)pi/w sin[(pi/w)theta] is an element ofH(2,q) with 2 < q < 1/(1-gimel(1)), where the numbers gimel(i) (i = 1, 2) satisfy 1/2 < &GIMEL;(1) < pi/w < &GIMEL;(2) < 1, the T-i are trigonometric vector functions, (chi) is a cutoff function, C-i (i = 1, 3) are constants, and r is the distance to the vertex. If D is convex, [u, sigma] is an element ofH(2,q) x H-2,H-q.