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The global regularity of vortex patches revisited

Joan Verdera

Abstract

We prove persistence of the regularity of the boundary of vortex patches for a large class of transport equations in the plane. The velocity field is given by convolution of the vorticity with an odd kernel, homogeneous of degree $-1$ and of class $C^2$ off the origin.

The global regularity of vortex patches revisited

Abstract

We prove persistence of the regularity of the boundary of vortex patches for a large class of transport equations in the plane. The velocity field is given by convolution of the vorticity with an odd kernel, homogeneous of degree and of class off the origin.

Paper Structure

This paper contains 5 sections, 5 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. The a priori estimate
  3. The logarithmic inequality
  4. End of the proof
  5. Appendix: local in time existence for the CDE

Key Result

Theorem 1

If $D_{0}$ is a bounded simply connected domain with boundary of class $C^{1+\gamma}$, $0<\gamma<1$, then there exists a weak solution of keq with initial condition $\chi_{D_0}$ of the form $\omega(z,t)=\chi_{D_{t}}(z)$ with $D_{t}$ a simply connected domain of class $C^{1+\gamma}$ for all times $t\

Theorems & Definitions (9)

  • Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof