On the stability of Lamb-Chaplygin dipole for the 2D Euler equation

TL;DR

This paper advances the understanding of the Lamb-Chaplygin dipole's stability in 2D Euler flows by establishing spectral stability without symmetry assumptions and refining orbital stability results.

Contribution

It proves spectral stability without symmetry conditions and introduces a new Lyapunov functional to improve orbital stability analysis.

Findings

Spectral stability of the linearized operator is established without symmetry assumptions.

A new coercive Lyapunov functional is constructed for stability analysis.

Quantitative bounds on fluctuations and velocity are derived.

Abstract

The Lamb-Chaplygin dipole is a traveling wave solution to the 2D incompressible Euler equation, whose orbital stability was established in [Abe-Choi, 2022] and [Abe-Choi-Jeong, 2025] assuming the odd symmetry in $x_2$ (O) and non-negativity in upper half-plane (N). This paper is devoted to further study of its stability in the following two aspects. Firstly, we prove the spectral stability of the linearized operator around the Lamb-Chaplygin dipole without conditions (O) or (N), based on the index theory established in [Lin-Zeng, 2022]. This excludes an instability mechanism by unstable eigenmodes, and provides rigorous evidence towards nonlinear stability in this general setting. Secondly, assuming (O) and (N), we refine the orbital stability results in [Abe-Choi, 2022] and [Abe-Choi-Jeong, 2025] quantitatively by proving a linear bound of the fluctuation and a uniform control of the moving velocity. Instead of using a variational approach, our proof relies on the construction of a new coercive Lyapunov functional with a delicate mixed structure: it is quadratic in the interior region, but linear in the exterior region.