9 - On the Fursikov approach to the moment problem for the three-dimensional Navier–Stokes equations

Summary

Abstract

In a series of papers Fursikov proposed a programme, based on analysing the moments of measure-valued statistical solutions, to obtain the density of initial data for which the corresponding solution of the 3D Navier– Stokes equations is regular. We illustrate the key points of the argument, and discuss some limitations of their method, by applying it to the case of the simple ODE ẋ +x−x² = 0 for which we know that such a density result cannot hold.

Introduction

Despite intensive efforts, the regularity problem for the three-dimensional Navier–Stokes equations is still unresolved, and is therefore very interesting to consider alternative approaches to this well-established problem. In this paper we discuss one such approach due to Fursikov (1984, 1986a & b, 1987; see also Vishik & Fursikov, 1988), which treats not the equations themselves, but rather the linear ‘Chain of Moments’ (CoM) equations that arise when one considers the moments of statistical (measure-valued) solutions. If successful, the Fursikov programme would show that the set of initial data that give rise to regular solutions of the three-dimensional Navier–Stokes equations forms a dense subset of some sufficiently regular Sobolev space.

One can define a statistical solution by taking a measure on the space of initial data and then letting this evolve in a natural way under the flow induced by the equations. The moments of the resulting time-dependent measure satisfy a chain of infinitely-many linear equations, the ‘Chain of Moments’ (Section 9.3.1), and with enough regularity this implication can be reversed (Section 9.3.3). Proving the existence of ‘regular’ solutions to these equations is complicated by the fact that solving for the kth moment requires knowledge of the (k+1)th moment. Therefore Fursikov adopts a more roundabout approach, based on considering the equations within a variational framework and minimising a functional that penalises non-regular solutions: in this way they show that the CoM equations have a regular solution for a dense set of initial moments (Section 9.4). The essential difficulty in deducing a result on the ‘density of regularity’ lies in relating these approximate moments to a positive measure on the space of initial data, which we discuss further in Section 9.5.