On the Two-Dimensional Navier–Stokes Equations in Stream Function Form

Abstract

In this paper the initial-boundary value problem of the Navier–Stokes equations in stream function form is considered. A trilinear form is introduced to deal with the nonlinear term. A weak formulation of this problem is provided. The existence of a weak solution is proved by an auxiliary semi-discrete Faedo–Galerkin scheme and a compactness argument. The uniqueness and regularity of the solution are discussed. Finally the convergence of the numerical solution and the converge rate with a certain choice of basis in the Faedo–Galerkin method are given.