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

TL;DR

This paper investigates the anisotropic regularity criteria for the 3D Navier-Stokes equations, showing that certain directional and Besov space norms of the velocity component blow up at the potential singularity time.

Contribution

It establishes new anisotropic regularity conditions involving directional and Besov space norms that must blow up if a finite-time singularity occurs in 3D Navier-Stokes solutions.

Findings

Blow-up of specific anisotropic norms at the singularity time.

Identification of directional regularity criteria linked to finite-time blow-up.

Extension of regularity criteria involving Besov spaces for Navier-Stokes.

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 $T^\star$.