On the critical one component regularity for 3-D Navier-Stokes system: general case

TL;DR

This paper investigates the conditions under which solutions to the 3D Navier-Stokes equations blow up, showing that certain directional Sobolev norms must become unbounded at the blow-up time.

Contribution

It establishes a new criterion linking finite-time blow-up to the divergence of directional Sobolev norms of the velocity component in the 3D Navier-Stokes system.

Findings

Directional Sobolev norms blow up at the blow-up time.

Blow-up is characterized by divergence in specific Sobolev spaces.

Provides a new perspective on regularity criteria for Navier-Stokes solutions.

Abstract

Let us consider an initial data $v_0$ for the homogeneous incompressible 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 $]4,\infty[$, and any unit vector $e$ of $\R^3$, the $L^p$ norm in time with value in $\dot{H}^{\frac 12+\frac 2 p }$ of $(v|e)_{\R^3}$ blows up at $T^\star$