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

Wang, Guodong

doi:10.1090/tran/8652

Nonlinear stability of planar steady Euler flows associated with semistable solutions of elliptic problems
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Trans. Amer. Math. Soc. 375 (2022), 5071-5095 Request permission

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 nondecreasing 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 improved 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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