Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

Abstract

Abstract We investigate the problem of classification of solutions for the steady Navier–Stokes equations in any cone-like domain. In the form of separated variables, u ( x , y ) = φ 1 ( r ) v 1 ( θ ) φ 2 ( r ) v 2 ( θ ) , where x = r cos θ and y = r sin θ in the polar coordinates, we obtain the expressions of all smooth solutions with C 0 Dirichlet boundary condition. In particular, we find some solutions which are Hölder continuous on the boundary but their gradients blow up at the corner, show that all solutions in the entire plane R 2 must be polynomials, and prove a sharp Liouville type theorem.