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 ${\dot{B}}_{p,q}^{-1+\frac{3}{p}}({\mathbb{R}}^3)$, with $3<p,q<\infty$, gives rise to a strong solution with a singularity at a finite time $T>0$, 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. (To appear in Communications in Mathematical Physics.)

36 pages

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