Blow-up of critical Besov norms at a potential Navier-Stokes singularity

Koch, Gabriel S, Gallagher, Isabelle and Planchon, Fabrice (2016) Blow-up of critical Besov norms at a potential Navier-Stokes singularity. Communications in Mathematical Physics, 343 (1). pp. 39-82. ISSN 0010-3616

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We show that the spatial norm of any strong Navier-Stokes solution in the space X must become unbounded near a singularity, where X may be any critical homogeneous Besov space in which local existence of strong solutions to the 3-d Navier-Stokes system is known. In particular, the regularity of these spaces can be arbitrarily close to -1, which is the lowest regularity of any Navier-Stokes critical space. This extends a well-known result of Escauriaza-Seregin-Sverak (2003) concerning the Lebesgue space $L^3$, a critical space with regularity 0 which is continuously embedded into the spaces we consider. We follow the "critical element" reductio ad absurdum method of Kenig-Merle based on profile decompositions, but due to the low regularity of the spaces considered we rely on an iterative algorithm to improve low-regularity bounds on solutions to bounds on a part of the solution in spaces with positive regularity.

Item Type: Article
Keywords: Navier-Stokes, regularity criteria, Besov spaces
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Gabriel Koch
Date Deposited: 15 Mar 2016 08:51
Last Modified: 08 Mar 2017 05:13

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