A Numerically Based Existence Theorem for the Navier-Stokes Equations

Abstract

We have recently experimented with a new spectral code for the spatially-periodic nonstationary Navier-Stokes equations, finding a plethora of apparently steady, periodic and chaotic solutions which seem interesting from the point of view of dynamical systems theory. This has stimulated our interest in the general problem of proving the existence of strict solutions of differential equations, that correspond to the computational results of numerical experiments. In this paper we analyze the numerical results of one of our experiments. The computational results appear to approximate what seems to be an unstable steady solution. Our objective is to provide a rigorous a-posteriori analysis, proving that there does indeed exist a corresponding close by strict solution, and that it is steady. Our method is based on showing that the computed solution satisfies criteria which imply the convergence of a fixed point iteration, in appropriate function spaces for an existence theorem, using the computed solution as the starting value