Construction of unstable concentrated solutions of the Euler and gSQG equations

Martin Donati

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arXiv · 2303.14657

Construction of unstable concentrated solutions of the Euler and gSQG equations

Martin Donati

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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 $\varepsilon$ to order $\varepsilon^\beta$ (with $\beta < 1$) in a time of order $|\ln\varepsilon|$. This proves in particular that the logarithmic lower-bound obtained in previous papers (in particular [P. Butt\`a 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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Martin Donati. 2023-03-26. Construction of unstable concentrated solutions of the Euler and gSQG equations. https://arxiv.org/abs/2303.14657

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