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

TL;DR

This paper proves that for Navier-Stokes solutions in critical Besov spaces, a finite-time singularity implies the unboundedness of the solution norm, extending previous results to a broader class of function spaces.

Contribution

It generalizes the blow-up norm unboundedness result to all critical Besov spaces where local existence is known, using profile decompositions and an iterative method.

Findings

Unbounded Besov norms at singularity time for Navier-Stokes solutions.

Extension of previous $L^3$ blow-up results to broader Besov spaces.

Introduction of an iterative technique applicable for large p values.

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 $3 <p,q< \infty$, 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.)