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 spaceḢ 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 "concentration-compactness" + "rigidity theorem" using "critical elements" 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. RésuméDans cet exposé, nous présentons un point de vue différent sur les études récentes concernant la régularité des solutions des équations de Navier-Stokes dans les espaces critiques. En particulier, nous démontrons que les solutions faibles qui restent bornées

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

“…See also the papers [7,13] for another approach to regularity using certain profile decompositions. It should be noticed that the uniqueness of u under condition (1.4) had been known earlier (see [15]).…”

Profile decompositions and applications to Navier-Stokes

“…The profile decompositions allow one to isolate the defects of compactness, thus allowing one to regain sufficient compactness to prove the existence of solutions enjoying important minimality properties. The results in [12] were made possible by the profile decompositions of P. Gérard [10] and I. Gallagher [6]. Recently in [17], the author adapted the method of S. Jaffard [11] to construct new profile decompositions for other spaces relevant to Navier-Stokes.…”

“…In the papers [12,9], we give an alternative proof of (R) L 3 (R 3 ) in the setting of "mild" solutions (locally smooth solutions satisfying an integral version of (NSE)). Our proof uses the powerful "critical element" method developed by Kenig and Merle for dispersive equations (where all solutions are "mild"), see e.g.…”

“…Recently in [17], the author adapted the method of S. Jaffard [11] to construct new profile decompositions for other spaces relevant to Navier-Stokes. Based on these, in collaboration with Gallagher and F. Planchon (see [9]) we give an alternative proof of the L 3,∞ regularity criterion of Escauriaza-Seregin-Šverák [5], a special case of which was accomplished in [12].…”

“…In the recent paper [12], the author and C. Kenig showed that one can apply a robust method from nonlinear dispersive (e.g. nonlinear Schrödinger) and hyperbolic (e.g.…”