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 $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any $p\in]4,\infty[,~q_1\in[1,2[,~\mu>0, ~q_2\in\bigl[2,\bigl(1/p+\mu\bigr)^{-1}\bigr[,~\kappa\in ]1,\infty[$, and any unit vector $e$, the $L^p$ estimate in time of $\bigl|(v(t)|e){\mathbb{R}^3}\bigr|{L^{\frac{3p}{p-2}}}^p +\bigl|(v(t)|e){\mathbb{R}^3}\bigr|^p{ \bigl(\dot{B}^{\mu+\frac2p+\frac2{q_1}-1}{q_1,\kappa}\bigr){\rm h} \bigl(\dot{B}^{\frac1{q_2}-\mu}{q_2,\kappa}\bigr){\rm v}}$ blows up at $T^\star$.