On the compactness of a nonlinear operator related to stream function-vorticity formulation for the Navier-Stokes equations

Abstract

A compactness proof of a nonlinear operator related to stream function-vorticity formulation for the Navier-Stokes equations is presented. The compactness of the operator provides important information for fixed-point formulations, especially for computer-assisted proofs based on Schauder's fixed-point theorem. Our idea for the compactness proof comes from books by Girault & Raviart and Ladyzhenskaia, and our principle would be also applied to convex polygonal regions.