Nonlinear stability of sinusoidal Euler flows on a flat two-torus

TL;DR

This paper proves the nonlinear stability of certain sinusoidal Euler flows on a flat torus, extending classical results by Arnold through spectral and isovortical analysis.

Contribution

It establishes nonlinear stability of least eigenfunction related sinusoidal flows under $L^p$ vorticity norms, improving classical energy-Casimir stability results.

Findings

Least eigenfunction sinusoidal flows are nonlinearly stable

Stability holds for all $1<p<+ +infty$ in $L^p$ vorticity norm

Distinguishing least eigenstates via isovortical property is key

Abstract

Sinusoidal flows are an important class of explicit stationary solutions of the two-dimensional incompressible Euler equations on a flat torus. For such flows, the steam functions are eigenfunctions of the negative Laplacian. In this paper, we prove that any sinusoidal flow related to some least eigenfunction is, up to phase translations, nonlinearly stable under $L^p$ norm of the vorticity for any $1<p<+\infty$, which improves a classical stability result by Arnold based on the energy-Casimir method. The key point of the proof is to distinguish least eigenstates with fixed amplitude from others by using isovortical property of the Euler equations.