Neocompact sets and stochastic Navier-Stokes equations

Summary

Abstract

We give a detailed exposition of the use of neocompact sets in proving existence of solutions to stochastic Navier-Stokes equations. These methods yield new results concerning optimality of solutions.

Introduction

In this paper we give a detailed exposition of the way in which the recent work of S. Fajardo and H. J. Keisler can be used to establish existence of solutions to stochastic Navier-Stokes equations. Fajardo & Keisler develop general methods for proving existence theorems in analysis, with the aim of embracing the many particular existence theorems that can be proved rather easily using nonstandard analysis. The machinery developed centres round the notion of a neocompact set – which is a weakening of the notion of a compact set of random variables with values in a metric space M - and the notion of a rich adapted probability space, in which any countable chain of nonempty neocompact sets has a nonempty intersection.

In the papers Capiński & Cutland used nonstandard methods to greatly simplify some known existence proofs for the deterministic Navier- Stokes equations and (using similar methods) solved a longstanding problem concerning existence of solutions to general stochastic Navier-Stokes equations. The aim here is to show how the main results of these papers can be obtained using the neocompactness methods developed in. In addition, these methods yield additional information concerning the nature of the set of solutions and existence of optimal solutions.