Critical one component anisotropic regularity for 3-D Navier-Stokes system

Abstract

Let us consider an initial data $v_0$ for the classical 3D Navier-Stokes equation with vorticity belonging to $L^{\frac{3}{2}}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^{\star }$, then for any $p\in ]4,\infty [,q_1\in [1,2[,\mu >0,q_2\in [2,(1/p+\mu {)}^{-1}[,\kappa \in ]1,\infty [$, and any unit vector $e$, the $L^p$ estimate in time of $\Vert (v(t)|e{)}_{{\mathbb{R}}^3}{\Vert }_{L^{\frac{3p}{p-2}}}^p+\Vert (v(t)|e{)}_{{\mathbb{R}}^3}{\Vert }_{\left({\dot{B}}_{q_1,\kappa }^{\mu +\frac{2}{p}+\frac{2}{q_1}-1}{)}_{\mathrm{h}}\right({\dot{B}}_{q_2,\kappa }^{\frac{1}{q_2}-\mu }{)}_{\mathrm{v}}}^p$ blows up at $T^{\star }$.

26 pages