A profile decomposition approach to the L∞/t (L3/ x) Navier–Stokes regularity criterion

Gallagher, Isabelle, Koch, Gabriel S and Planchon, Fabrice (2013) A profile decomposition approach to the L∞/t (L3/ x) Navier–Stokes regularity criterion. Mathematische Annalen, 355 (4). pp. 1527-1559. ISSN 0025-5831

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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 Kenig and Koch (Ann Inst H Poincaré Anal Non Linéaire 28(2):159–187, 2011). Specifically, we prove that strong solutions which remain bounded in the space L3(R3) do not become singular in finite time, a known result established by Escauriaza et al. (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).

Item Type: Article
Additional Information: Published online 12 July 2012, online ISSN 1432-1807
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Related URLs:
Depositing User: Gabriel Koch
Date Deposited: 14 Nov 2012 12:25
Last Modified: 14 Mar 2017 02:18
URI: http://sro.sussex.ac.uk/id/eprint/41953

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