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

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 Ḣ 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. Šverák using a different approach. We use the method of “concentrationcompactness” + “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. Introduction In recent studies, the idea of establishing the existence of so-called “critical elements” (or the earlier “minimal blow-up solutions”) has led to significant progress in the theory of “critical” dispersive and hyperbolic equations such as the energy-critical nonlinear Schrödinger equation [2, 3, 8, 26, 45, 53], masscritical nonlinear Schrödinger equation [29, 30, 31, 50, 51], Ḣ 1 2 -critical nonlinear Schrödinger equation [25], energy-critical nonlinear wave equation [27], energycritical and mass-critical Hartree equations [38, 39, 40, 41, 42] and energy-critical wave maps [9, 34, 48, 49]. In this paper we exhibit the generality of the method of “critical elements” by applying it to a parabolic system, namely the standard Navier-Stokes equations (NSE): (0.1) ut −∆u+ (u · ∇)u +∇p = 0 ∇ · u = 0 Department of Mathematics, University of Chicago; Chicago, Il 60637, USA; [email protected]; supported in part by NSF grant DMS-0456583 Department of Mathematics, University of Chicago; Chicago, Il 60637, USA; [email protected] For simplicity, we have set the coefficient of kinematic viscosity ν = 1.