For the vorticity–velocity–pressure form of the Navier–Stokes equations on a bounded plane domain with corners

Abstract

We show existence and regularity for the boundary value problems of the Navier–Stokes equations with non-standard BCs on a bounded plane domain with non-convex corners. We assign the vorticity value ω = ω 0 and the velocity normal component u ⋅ n = u 0 ⋅ n over the non-convex corner, the dynamic pressure value p + | u | 2 / 2 = p 0 over inflow and outflow boundaries, and so on. We construct the corner singularity functions for the Stokes operator with zero vorticity and velocity normal component BCs, subtract its leading singularity from the solution by defining the coefficient of the singularity and show increased regularity for the remainder. The solution is determined by the smoother part and the coefficients of the singularities. It is seen from the singularity that the dynamic pressure has a transition layer that changes the sign (at θ = π / 2 in the domain). The obtained results can be applied to general polygonal domains and the cavity flows.