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A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang

Qi S. Zhang

Abstract

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail. In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.

A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang

Abstract

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail. In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.

Paper Structure

This paper contains 13 sections, 41 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Regularity Criterion
  3. Critical and slightly super critical regularity conditions
  4. Criticality of ASNS and closing of the scaling gap
  5. Ancient Solutions
  6. Stationary Solutions
  7. D solutions
  8. Self-similar solutions
  9. Addendum: Extra results on D solutions, by Xinghong Pan and Q. S. Zhang
  10. Proof of Theorem \ref{['thutheta']}, one component axially symmetric periodic solutions
  11. Proof of Theorem \ref{['thmexpanding']}, axially symmetric solutions on aperture domains
  12. Proof of case (i): $0\leq \alpha <1/2$.
  13. Proof of case (ii): $1/2\leq \alpha <3/4$.

Key Result

Theorem 2.1

(LZ11) Let $v=v(x, t)$ be a Leray-Hopf solution to (eqasns) in the space time region ${{\mathbb R}}^3 \times [0, T]$. Assume that the initial value satisfies, $|r v^\theta(x, 0)|<C$. Suppose also $v(\cdot, t) = \nabla \times B(\cdot, t)$ with $\sup_{0 < t < T}\|B(\cdot, t)\|_{{\rm BMO}} \leq C_\ast$

Theorems & Definitions (47)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 37 more