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The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

Marc Magaña, Joan Mateu, Joan Orobitg

Abstract

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as $\varepsilon \to 0$ in little Hölder spaces.

The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

Abstract

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius , which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as in little Hölder spaces.
Paper Structure (14 sections, 25 theorems, 186 equations)

This paper contains 14 sections, 25 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modified Bessel functions
  4. Inequalities for Hölder continuous functions and Kernel estimates
  5. Existence and uniqueness of regular solutions
  6. Local-in-time existence of solutions
  7. Global existence of smooth solutions
  8. Weak solutions to the QGSW equation
  9. Existence of weak solutions
  10. Uniqueness of weak solutions
  11. Vortex patches
  12. Local existence and uniqueness of solutions to the CDE
  13. Global regularity for vortex patches
  14. Convergence to the 2D Euler Equations

Key Result

Theorem 1.1

Let $\Omega_0$ be a bounded domain with boundary of class $C^{1,\gamma},\;0<\gamma<1.$ Then the has a unique weak solution of the form $q(x,t)=\chi_{\Omega_t}(x),$ with $\Omega_t$ a bounded domain with boundary of class $C^{1,\gamma}.$

Theorems & Definitions (41)

  • Theorem 1.1
  • Definition 1
  • Lemma 2.1
  • Definition 2
  • Lemma 2.2
  • proof
  • Remark 1
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • ...and 31 more