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Boundary regularity of uniformly rotating vortex patches and an unstable elliptic free boundary problem

Yuchen Wang, Guanghui Zhang, Maolin Zhou

Abstract

In this paper, we consider a sign-changing free boundary problem that comes from the boundary regularity of rotating vortex patches of the two-dimensional incompressible Euler equations. The complete classification of singular points has been obtained through establishing a new Weiss-type monotonicity formula. Upon these results, we prove that only $90^\circ$ corner type of singularity could happen at the boundary of a Lipschitz rotating vortex patch, while the other parts are $C^\infty$ smooth.

Boundary regularity of uniformly rotating vortex patches and an unstable elliptic free boundary problem

Abstract

In this paper, we consider a sign-changing free boundary problem that comes from the boundary regularity of rotating vortex patches of the two-dimensional incompressible Euler equations. The complete classification of singular points has been obtained through establishing a new Weiss-type monotonicity formula. Upon these results, we prove that only corner type of singularity could happen at the boundary of a Lipschitz rotating vortex patch, while the other parts are smooth.
Paper Structure (5 sections, 11 theorems, 115 equations, 1 figure)

This paper contains 5 sections, 11 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A sign-changing two phases unstable elliptic free boundary problem: Weiss-type monotonicity formula and classification of blow-up limits
  3. Singularities of the two-dimensional unstable elliptic free boundary problem
  4. Blow-up analysis
  5. Uniform regularity

Key Result

Theorem 1.1

Suppose $D$ is a bounded domain, $u$ solves the 2-dim unstable elliptic free boundary problem and is positive in $D$ The free boundary $\partial D$ is locally real-analytic up to the singular sets which consist of For more accurate descriptions on singular points, please see Proposition P:Class.

Figures (1)

  • Figure 1: Classification on the singular points

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Lemma \ref{['L:Mono-F']}
  • Proposition 2.3: Classification of blow-up limits
  • Remark 2.4
  • ...and 12 more