Muliti-scale regularity of axisymmetric Navier-Stokes equations

TL;DR

This paper establishes multi-scale regularity criteria for the swirl component in 3D axisymmetric Navier-Stokes equations, enabling solution continuation beyond a certain time under specific integrability conditions.

Contribution

It introduces new multi-scale regularity criteria for the swirl component, extending the understanding of solution continuation in axisymmetric Navier-Stokes equations.

Findings

Solution can be extended beyond time T if swirl component satisfies certain integrability conditions.

Provides a priori estimates for equations involving (,) variables.

Establishes criteria involving mixed Lebesgue spaces for regularity.

Abstract

By applying the delicate \textit{a priori} estimates for the equations of $(\Phi,\Gamma)$, which is introduced in the previous work, we obtain some multi-scale regularity criteria of the swirl component $u^{\theta}$ for the 3D axisymmetric Navier-Stokes equations. In particularly, the solution $\mathbf{u}$ can be continued beyond the time $T$, provided that $u^{\theta}$ satiesfies $$ u^{\theta} \in L^{p}_{T}L^{q_{v}}_{v}L^{q_{h},w}_{h},~~\frac{2}{p}+\frac{1}{q_{v}}+\frac{2}{q_{h}}\leq 1, ~2<q_{h}\leq\infty,~\frac{1}{q_{v}}+\frac{2}{q_{h}}<1. $$