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

TL;DR

This paper establishes the orbital stability of certain exact steady solutions of the 2D incompressible Euler equations in a disk, including the truncated Lamb dipole, using variational methods and conserved quantities.

Contribution

It provides a sharp stability proof for solutions expressed with Bessel functions, notably including the truncated Lamb dipole, advancing understanding of vortex stability in bounded domains.

Findings

Proved orbital stability of solutions involving Bessel functions in a disk.

Identified the truncated Lamb dipole as a special case with stable behavior.

Developed a variational characterization based on conserved quantities.

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.