A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion

TL;DR

This paper develops a profile decomposition method for Navier-Stokes equations in critical spaces, providing a new proof that solutions bounded in $L^3$ do not develop singularities, and explores minimal norm initial data for potential singularities.

Contribution

It introduces a profile decomposition approach in critical Besov spaces for Navier-Stokes and extends minimal norm results for initial data leading to singularities.

Findings

Solutions bounded in $L^3$ do not become singular in finite time.

Developed a profile decomposition in critical Besov spaces.

Existence of minimal norm initial data for potential singularities.

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 Poincar\'e Anal Non Lin\'eaire 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).