8 - The regularity problem for the three-dimensional Navier–Stokes equations

Summary

Abstract

This paper gives a brief summary of some of the main results concerning the regularity of solutions of the three-dimensional Navier–Stokes equations. We then outline the basis of a numerical algorithm that, at least in theory, can verify regularity for all initial conditions in any bounded subset of H¹.

Introduction

The aim of this paper is to present some partial results concerning the problem of regularity of global solutions of the three-dimensional Navier–Stokes equations. Since these equations form the fundamental model of hydrodynamics it is a matter of great importance whether or not they can be uniquely solved. However, one hundred and fifty years after the Navier–Stokes model was presented for the first time, we still lack an existence and uniqueness theorem, and the most significant contributions to the subject remain those of Leray (1934) and Hopf (1951).

Nevertheless, there have been many advances since their work, and it would not be possible to give an exhaustive presentation of these in a short article. We give a brief overview of some of the main results, and then concentrate on one specific and in some ways non-standard approach to the problem, with a discussion of the feasibility of testing for regularity via numerical computations following Chernysehnko et al. (2007), Dashti & Robinson (2008), and Robinson & Sadowski (2008). In some ways this contribution can be viewed as a companion to the introductory review by Robinson (2006).