About some possible blow-up conditions for the 3-D Navier-Stokes equations

Abstract

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from \cite{Isab}, we prove that if one component of the velocity remains small enough in a sub-space of ${\dot{H}}^{\frac{1}{2}}$ "almost" scaling invariant, then the 3D Navier Stokes is globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form $L_T^p({\dot{B}}_{2,\infty }^{\alpha ,\frac{2}{p}-\frac{1}{2}-\alpha })$ or $L_T^p({\dot{B}}_{q,\infty }^{\frac{2}{p}+\frac{3}{q}-2})$.