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Stability of a class of exact solutions of the incompressible Euler equation in a disk

Guodong Wang

Abstract

We prove a sharp orbital stability result for a class of exact steady solutions, expressed in terms of Bessel functions of the first kind, of the two-dimensional incompressible Euler equation in a disk. A special case of these solutions is the truncated Lamb dipole, whose stream function corresponds to the second eigenfunction of the Dirichlet Laplacian. The proof is achieved by establishing a suitable variational characterization for these solutions via conserved quantities of the Euler equation and employing a compactness argument.

Stability of a class of exact solutions of the incompressible Euler equation in a disk

Abstract

We prove a sharp orbital stability result for a class of exact steady solutions, expressed in terms of Bessel functions of the first kind, of the two-dimensional incompressible Euler equation in a disk. A special case of these solutions is the truncated Lamb dipole, whose stream function corresponds to the second eigenfunction of the Dirichlet Laplacian. The proof is achieved by establishing a suitable variational characterization for these solutions via conserved quantities of the Euler equation and employing a compactness argument.
Paper Structure (6 sections, 21 theorems, 115 equations)

This paper contains 6 sections, 21 theorems, 115 equations.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. An energy-enstrophy inequality
  4. Proof of Theorem \ref{['thmmain2']}
  5. Proof of Theorem \ref{['thmmain']}
  6. Rotating solutions

Key Result

Lemma 1.1

The space $\mathbf V_{n,k}$ can be characterized by As a consequence, any element in $\mathbf V_{n,k}$ is a steady solution of vor.

Theorems & Definitions (39)

  • Lemma 1.1
  • proof
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1: Bo
  • Lemma 2.2
  • Lemma 2.3: BMA
  • Lemma 2.4: BACT
  • ...and 29 more