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Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions

Evan Miller

Abstract

In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler equation in four dimensions required the additional assumption that $\frac{ω^0}{r^2}\in L^\infty$, which can fail even for Schwartz class solutions. The key advance is a new bound on the vortex stretching term that only requires $\frac{ω^0}{r^2}\in L^{2,1}(\mathbb{R}^4)$, which is generically true for any axisymmetric, swirl-free initial data $u^0\in H^s\left(\mathbb{R}^4\right), s>4$, with reasonable decay at infinity.

Global regularity for axisymmetric, swirl-free solutions of the Euler equation in four dimensions

Abstract

In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler equation in four dimensions required the additional assumption that , which can fail even for Schwartz class solutions. The key advance is a new bound on the vortex stretching term that only requires , which is generically true for any axisymmetric, swirl-free initial data , with reasonable decay at infinity.
Paper Structure (5 sections, 12 theorems, 79 equations)

This paper contains 5 sections, 12 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Bound on vortex stretching
  4. Global regularity
  5. Hypothesis satisfied for generic data

Key Result

Theorem 1.1

Suppose $u^0 \in H^s_{df}\left(\mathbb{R}^4\right), s>4$ is axisymmetric, swirl-free and that $\frac{\omega^0}{r^2}\in L^{2,1}\left(\mathbb{R}^4\right)$. Then there exists a global smooth solution of the Euler equation $u\in C\left([0,+\infty); H^s\left(\mathbb{R}^4 \right)\right)$, and for all $0<t where $C>0$ is an absolute constant independent of $u^0$. Furthermore, the condition $\frac{\omega^

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 18 more