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Mathematics > Analysis of PDEs

arXiv:2210.01405 (math)
[Submitted on 4 Oct 2022 (v1), last revised 9 Oct 2022 (this version, v2)]

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

Authors:Guodong Wang, Bijun Zuo
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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.
Comments: Biot-Savart law is corrected
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:2210.01405 [math.AP]
  (or arXiv:2210.01405v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.2210.01405

Submission history

From: Guodong Wang [view email]
[v1] Tue, 4 Oct 2022 06:26:02 UTC (17 KB)
[v2] Sun, 9 Oct 2022 02:39:24 UTC (18 KB)
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