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

2021/07/01 by Guodong Wang, Wang, Guodong · 1 citation
Mathematics · #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2107.00217

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.

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