Stability of 2D steady Euler flows related to least energy solutions of the Lane-Emden equation

TL;DR

This paper establishes the nonlinear stability of certain 2D steady Euler flows linked to least energy solutions of the Lane-Emden equation, using variational methods and conserved quantities.

Contribution

It introduces a new variational characterization of least energy solutions and proves their orbital stability in various norms for a broad class of domains.

Findings

Orbital stability of flows in $L^s$ and energy norms.

Stability holds for a large class of domains and exponents.

New variational approach based on vorticity characterization.

Abstract

In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the $L^s$ norm of the vorticity for any $s\in(1,+\infty)$ and the energy norm. As a consequence, nonlinear stability is obtained when the least energy solution is unique, which actually holds for a large class of domains and exponents. The proofs are based on a new variational characterization of least energy solutions in terms of the vorticity, a compactness argument, and proper use of conserved quantities of the Euler equation.