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

Abstract : We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 0$, then the norm of the solution in that Besov space becomes unbounded at time $T$. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza, Seregin and Sverak (Uspekhi Mat. Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in $L^3(\mathbb{R}^3)$. Our proof uses profile decompositions and is based on our previous work (Math. Ann. 355(4):1527--1559, 2013) which provided an alternative proof of the $L^3(\mathbb{R}^3)$ result. For very large values of $p$, an iterative method, which may be of independent interest, enables us to use some techniques from the $L^3(\mathbb{R}^3)$ setting.
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Contributor : Jean-Louis Thomin <>
Submitted on : Friday, November 20, 2015 - 2:29:23 PM
Last modification on : Wednesday, December 9, 2020 - 3:08:11 PM

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Isabelle Gallagher, Gabriel S. Koch, Fabrice Planchon. Blow-up of critical Besov norms at a potential Navier-Stokes singularity. Communications in Mathematical Physics, Springer Verlag, 2016, 343 (1), pp.39-82. ⟨10.1007/s00220-016-2593-z⟩. ⟨hal-01231551⟩



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