Construction of unstable concentrated solutions of the Euler and gSQG equations

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.