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

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