An alternative approach to regularity for the Navier-Stokes equations in critical spaces

TL;DR

This paper introduces a novel application of the concentration-compactness and rigidity method to study regularity of Navier-Stokes solutions in critical spaces, specifically showing bounded solutions in ot H^{1/2} do not become singular.

Contribution

It applies the concentration-compactness and rigidity method, previously used for dispersive equations, to a parabolic Navier-Stokes problem, providing a new proof of regularity in critical spaces.

Findings

Bounded mild solutions in ot H^{1/2} do not develop singularities.

First application of this method to a parabolic PDE.

Technical restrictions limit the current scope, with plans for broader cases.

Abstract

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. We remark that we have restricted our attention to a special case due only to a technical restriction, and plan to return to the general case (the $L^3$ setting) in a future publication.