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

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 H˙12 do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Šverák using a different approach. We use the method of “concentration-compactness” + “rigidity theorem” using “critical elements” which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authorsʼ knowledge, this is the first instance in which this method has been applied to a parabolic equation.