Construction of unstable concentrated solutions of the Euler and gSQG equations

Martin Donati

doi:10.3934/dcds.2024053

Article Contents

2024, Volume 44, Issue 10: 3109-3134. Doi: 10.3934/dcds.2024053

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Construction of unstable concentrated solutions of the Euler and gSQG equations

Martin Donati^{1, 2,}

1.
Université de Lyon, ENS de Lyon, Unité de Mathématiques Pures et Appliquées, 69364 Lyon Cedex 07, France
2.
Université Grenoble Alpes, Institut Fourier, F-38000 Grenoble, France

Received: July 2023

Revised: April 2024

Early access: April 25, 2024

Published online: April 25, 2024

The author is supported by the Simons Collaboration on Wave Turbulence.

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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| $. In particular, this proves 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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Mathematics Subject Classification: Primary: 76B47, 34A34.

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Figure 1. The domain $ \Omega_{\frac{3}{4}} $

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Figure 2. Plot of the domain $ S_\delta(D) $ for left to right, $ \delta = \frac{2}{5} $, $ \delta = \frac{2}{3} $, and $ \delta = \frac{3}{4} $

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Figure 3. Plot of the map $ r_{S_\delta(D)} \circ S_\delta : x \mapsto |S_\delta'(x)|(1-|x|^2) $ for, left to right, a stable case ($ \delta = \frac{2}{5} $), the critical case ($ \delta = \frac{2}{3}) $, and an unstable case ($ \delta = \frac{9}{10} $)

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Figure 4. Plot of $ g(\alpha) $ in the range $ \alpha \in [1,2) $

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Figure 5. Plot of $ h(\alpha) $ in the range $ \alpha \in [1,2) $, with $ N = 7 $ (left) and $ N = 9 $ (right)

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