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Sharp local well-posedness of $C^1$ vortex patches

Seungjae Lee

Abstract

It is well known that the boundary dynamics of vortex patches is globally well-posed in the Hölder space $C^{1,α}$ for $0<α<1$, whereas the well-posedness in $C^1$ remains an open problem, even locally. In this paper, we establish the local well-posedness for vortex patches in the space $C^{1,\varphi}$ defined via a modulus of continuity $\varphi$ that satisfies certain structural assumptions. Our class includes curves that are strictly rougher than the Hölder-continuous ones, with prototypical examples being $\varphi(r) = (-\log r)^{-s}$ for $s>3$. Motivated by the fact that the velocity operator in the contour dynamics equation is a nonlinear variant of the Hilbert transform, we study the system of equations satisfied by the curve parametrization $γ\in C^{1,\varphi}$ and its Hilbert transform. In doing so, we derive several properties of the Hilbert transform and its variants in critical spaces, which are essential for controlling the velocity operator and its Hilbert transform.

Sharp local well-posedness of $C^1$ vortex patches

Abstract

It is well known that the boundary dynamics of vortex patches is globally well-posed in the Hölder space for , whereas the well-posedness in remains an open problem, even locally. In this paper, we establish the local well-posedness for vortex patches in the space defined via a modulus of continuity that satisfies certain structural assumptions. Our class includes curves that are strictly rougher than the Hölder-continuous ones, with prototypical examples being for . Motivated by the fact that the velocity operator in the contour dynamics equation is a nonlinear variant of the Hilbert transform, we study the system of equations satisfied by the curve parametrization and its Hilbert transform. In doing so, we derive several properties of the Hilbert transform and its variants in critical spaces, which are essential for controlling the velocity operator and its Hilbert transform.

Paper Structure

This paper contains 23 sections, 13 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Vortex Patches
  3. Main Results
  4. Historical Backgrounds
  5. Ideas of Proof
  6. Organization
  7. Preliminaries
  8. Notations
  9. Hilbert Transform and Its Properties
  10. Local Well-Posedness of Vortex Patches
  11. Functional Setup
  12. Regularity of the Velocity Field and Its Hilbert Transform
  13. Lipschitz Estimates for the Velocity Field and Its Hilbert Transform
  14. Proof of Theorem \ref{['Thm2']}
  15. Proofs of Lemmas
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\varphi$ be a modulus of continuity satisfying assumptions A1--A3 and $\gamma_0, \mathcal{H}[\gamma_0] \in C^{1,\varphi}(\mathbb T)$. We further assume that $\gamma$ is a non-degenerate parametrization: Then there exists $T = T(\varphi, \Vert\gamma_0\Vert_{C^{1,\varphi}}, \Vert\mathcal{H}[\gamma_0]\Vert_{C^{1,\varphi}},|\gamma_0|_\star)$ and the unique solution $(\gamma(t),\mathcal{H}[\gamma

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Remark 1: Remarks on the Main Theorem
  • Remark 2: Remarks on the assumptions on moduli
  • Remark 3
  • Lemma 1: $C^{\widetilde{\varphi}}$ estimate of $\mathcal{H}[f]$
  • Lemma 2: An Algebra Property of the Hilbert Transform
  • Remark 4
  • Lemma 3: Estimate of $\mathcal{H}[f^n]$
  • Remark 5
  • ...and 20 more