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arXiv · 2401.10496

On the locally self-similar blowup for the generalized SQG equation

Anne Bronzi·Ricardo Guimarães·Cecilia Mondaini
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Abstract

We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in $\mathbb{R}^2$. Under an $L^r$ growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its $L^p$ asymptotic behavior can be characterized, for suitable $r, p$. Our results extend the work [Xue; Journal of Differential Equations, 2016] regarding the SQG equation, and also partially recover the results proved in [Cannone, Xue; Proceedings of the American Mathematical Society, 2015] concerning globally self-similar solutions of the gSQG equation.

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Anne Bronzi, Ricardo Guimarães, Cecilia Mondaini. 2024-01-19. On the locally self-similar blowup for the generalized SQG equation. https://arxiv.org/abs/2401.10496

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