University of Cambridge > Talks.cam > Geometric Analysis and Partial Differential Equations seminar > Blow-up of arbitrarily rough critical Besov norms at any Navier-Stokes singularity

Blow-up of arbitrarily rough critical Besov norms at any Navier-Stokes singularity

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We show that the spatial norm in any critical homogeneous Besov space in which local existence of strong solutions to the 3-d Navier-Stokes equations is known must become unbounded near a singularity. 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. This is joint work with I. Gallagher (Paris 7) and F. Planchon (Nice).

This talk is part of the Geometric Analysis and Partial Differential Equations seminar series.

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