Construction of unstable concentrated solutions of the Euler and gSQG equations

2023/03/26 by Martin Donati, Donati, Martin · 1 citation
Earth and Planetary Sciences · Physics and Astronomy · #34A34 #76B47 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2303.14657

openalex publication_date 2023/03/26 · openalex created_date 2023/03/31 · openalex updated_date 2026/07/28

Abstract

In this paper we construct solutions to the Euler and gSQG equations that are concentrated near unstable stationary configurations of point-vortices. Those solutions are themselves unstable, in the sense that their localization radius grows from order ε to order ε^β (with β< 1) in a time of order |lnε|. This proves in particular that the logarithmic lower-bound obtained in previous papers (in particular [P. Buttà and C. Marchioro, Long time evolution of concentrated Euler flows with planar symmetry, SIAM J. Math. Anal., 50(1):735-760, 2018]) about vorticity localization in Euler and gSQG equations is optimal. In addition we construct unstable solutions of the Euler equations in bounded domains concentrated around a single unstable stationary point. To achieve this we construct a domain whose Robin's function has a saddle point.

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