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arXiv · 2107.00217

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

Guodong Wang
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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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Guodong Wang. 2021-07-01. Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems. https://arxiv.org/abs/2107.00217

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