A profile decomposition approach to the $L_t^{\infty }(L_x^3)$ Navier-Stokes regularity criterion

Abstract

In this paper we continue to develop an alternative viewpoint on recent studies of Navier-Stokes regularity in critical spaces, a program which was started in the recent work by C. Kenig and the second author (Ann Inst H Poincaré Anal Non Linéaire 28(2):159-187, 2011). Specifically, we prove that strong solutions which remain bounded in the space $L^3(R^3)$ do not become singular in finite time, a known result established by Escauriaza, Seregin and Sverak (Uspekhi Mat Nauk 58(2(350)):3-44, 2003) in the context of suitable weak solutions. Here, we use the method of "critical elements" which was recently developed by Kenig and Merle to treat critical dispersive equations. Our main tool is a "profile decomposition" for the Navier-Stokes equations in critical Besov spaces which we develop here. As a byproduct of this tool, assuming a singularity-producing initial datum for Navier-Stokes exists in a critical Lebesgue or Besov space, we show there is one with minimal norm, generalizing a result of Rusin and Sverak (J Funct Anal 260(3):879-891, 2011).

35 pages; Mathematische Annalen 2012

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• Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces
• Uniform stabilization of Navier-Stokes equations in critical $L^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls
• Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations
• Minimal initial data for potential Navier-Stokes singularities
• Uniform stabilization of 3D Navier-Stokes equations in critical Besov spaces with finite dimensional, tangential-like boundary, localized feedback controllers
• About some possible blow-up conditions for the 3-D Navier-Stokes equations
• Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations
• The lack of compactness in the Sobolev-Strichartz inequalities
• Some remarks about the possible blow-up for the Navier-Stokes equations