Some Remarks on Regularity Criteria of Axially Symmetric Navier-Stokes Equations

Abstract

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^{\theta }$ of the velocity. Second, the precise Green function for the operator $-(\Delta -\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^{\theta }$ of the vorticity.

arXiv commentFinal version, to appear in Comm. Pure Appl. Anal

References in corpus (4)

• Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II
• Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations
• Regularity of Solution to Axis-symmetric Navier-Stokes Equations with a Slightly Supercritical Condition
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Cited by in corpus (5)

• On the Vanishing of some D-solutions to the Stationary Magnetohydrodynamics System
• A Note On Local Regularity of Axisymmetric Solutions to the Navier-Stokes Equations
• A Slightly Supercritical Condition of Regularity of Axisymmetric Solutions to the Navier-Stokes Equations
• Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations
• A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang