Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments

Abstract

The solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. The corner singularity theory for the solutions of elliptic boundary value problems on domains with corners or edges has been well established in the past century and also in recent years. The corner singularity functions provide an appropriate mathematical structure to understand the physical trajectories of the fluid particles. It has been investigated for general elliptic boundary value problems and also extended to some non-elliptic problems. Currently, the theory has been constructed for compressible viscous Stokes and NavierStokes systems on polygonal and polyhedral domains to analyze the structure of the solution near the corners and edges. Several interesting results about the regularity of the solution cannot be extended if one of the following situations appears: The domain has corners, edges and cusp, etc. On the boundary, change of boundary conditions at some points, discontinuities of the solutions, and singularities of the coefficients. This article reviewed the structure of the solution and regularity results of the stationary Stokes and Navier-Stokes equations on polygonal domains with convex or non-convex corners.