Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces

Abstract

We obtain an improved blow-up criterion for solutions of the Navier-Stokes equations in critical Besov spaces. If a mild solution $u$ has maximal existence time $T^{\ast }<\infty$, then the non-endpoint critical Besov norms must become infinite at the blow-up time: $$\underset{t\uparrow T^{\ast }}{\lim }\Vert u(\cdot ,t){\Vert }_{{\dot{B}}_{p,q}^{-1+3/p}({\mathbb{R}}^3)}=\infty ,3<p,q<\infty .$$ In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.

To appear in Analysis & PDE

References in corpus (3)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Blow-up of critical Besov norms at a potential Navier-Stokes singularity
• Uniqueness Results for Weak Leray-Hopf Solutions of the Navier-Stokes System with Initial Values in Critical Spaces

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