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

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