On the locally self-similar blowup for the generalized SQG equation
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arXiv
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| Format: | Preprint |
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2024
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| author | Bronzi, Anne Guimarães, Ricardo Mondaini, Cecilia |
| author_facet | Bronzi, Anne Guimarães, Ricardo Mondaini, Cecilia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the locally self-similar blowup for the generalized SQG equation Bronzi, Anne Guimarães, Ricardo Mondaini, Cecilia Analysis of PDEs 35C06, 35B44, 35Q35, 35R11, 35Q86 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. |
| title | On the locally self-similar blowup for the generalized SQG equation |
| topic | Analysis of PDEs 35C06, 35B44, 35Q35, 35R11, 35Q86 |
| url | https://arxiv.org/abs/2401.10496 |