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 × H1-regularity. Precisely, near a non-convex vertex located at (0, 0), [u, p] = 𝒞1[Φ1, φ1] + 𝒞2[Φ2, φ2] + [u R , p R ], [u R , p R ] ∈ H s × H s−1 for s ∈ (λ2 + 1, 2], where Φ i = χ r λ i 𝒯 i (θ), φ i = χ r λ i −1ξ i (θ) with 1/2 < λ1 < λ2 < 1, χ is a smooth cutoff function, and 𝒞 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.