Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems

TL;DR

This paper establishes the nonlinear stability of certain steady Euler flows in two dimensions by linking their stability to semistable solutions of elliptic problems, extending classical stability results.

Contribution

It introduces a new stability criterion for steady Euler flows based on semistable elliptic solutions, extending Arnol'd's second stability theorem.

Findings

Proves nonlinear stability in L^p norm for flows with semistable stream functions.

Shows such flows have strict local energy maxima among rearranged vorticities.

Extends classical stability theorems to broader classes of flows.

Abstract

This paper is devoted to the study of nonlinear stability of steady incompressible Euler flows in two dimensions. We prove that a steady Euler flow is nonlinearly stable in $L^p$ norm of the vorticity if its stream function is a semistable solution of some semilinear elliptic problem with strictly increasing nonlinearity. The idea of the proof is to show that such a flow has strict local maximum energy among flows whose vorticities are rearrangements of a given function, with the help of an improvement version of Wolansky and Ghil's stability theorem. The result can be regarded as an extension of Arnol'd's second stability theorem.