Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II

TL;DR

This paper establishes regularity of axisymmetric Navier-Stokes solutions with swirl under specific growth conditions, providing lower bounds on blow-up rates and extending understanding of singularity formation in fluid dynamics.

Contribution

It proves regularity at a critical time for solutions satisfying certain growth bounds, advancing the analysis of potential singularities in axisymmetric Navier-Stokes flows.

Findings

Solutions are regular at time zero under specified growth conditions.

Growth bounds prevent finite-time blow-up of solutions.

Extends criteria for regularity in axisymmetric Navier-Stokes equations.

Abstract

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}} $ or, for some $\e > 0$, $|v (x,t)| \le C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero.