Blow-up of critical norms for the 3-D Navier-Stokes equations

Abstract

Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in ${\mathbb{R}}^3\times [0,T)$. It was proved that if $u_3\in L^{\infty }(0,T;{\dot{B}}_{p,q}^{-1+3/p}({\mathbb{R}}^3))$ for $3<p,q<\infty$ and $u_h\in L^{\infty }(0,T;BM{O}^{-1}({\mathbb{R}}^3))$ with $u_h(T)\in VM{O}^{-1}({\mathbb{R}}^3)$, then $u$ can be extended beyond $T$. This result generalizes the recent result proved by Gallagher, Koch and Planchon, which requires $u\in L^{\infty }(0,T;{\dot{B}}_{p,q}^{-1+3/p}({\mathbb{R}}^3))$. Our proof is based on a new interior regularity criterion in terms of one velocity component.

References in corpus (2)

• Blow-up of critical Besov norms at a potential Navier-Stokes singularity
• On the critical one component regularity for 3-D Navier-Stokes system: general case

Cited by in corpus (2)

• On blowup of nonendpoint borderline Lorentz norms for the Navier-Stokes equations
• Local boundary regularity for the Navier-Stokes equations in nonendpoint borderline Lorentz spaces