The Stationary Navier-Stokes System with No-Slip Boundary Condition on Polygons: Corner Singularity and Regularity

Abstract

We study the stationary Navier-Stokes system with no-slip boundary condition on polygonal domains. Near each non-convex vertex the solution is shown solution. For showing the regularity we apply the Mellin transform to to have a unique decomposition by singular and regular parts. The singular part is defined by a linear combination of the corner singularity functions of the Stokes type and the regular part is shown to have the H (2)xH(1)-regularity. Precisely, near a non-convex vertex located at (0, 0), [u, p]=?(1)[phi(1), phi(1)] + ?(2)[phi(2), phi(2)] + [u (R), p(R)], [u (R), p(R)]H (s)xH(s-1) for s((2)+1, 2], where phi(i)=r(i) (i)(), phi(i)=r(i)-1(i)() with 1/2<(1)<(2)<1, is a smooth cutoff function, and ?(i) is the stress intensity factor. Hence the velocity vector is not Lipshitz continuous at non-convex vertices and the pressure value is infinite there. Also viscous stress tensor and vorticity values blow up near non-convex vertices.