Existence of hyperbolic blow-up to the generalized quasi-geostrophic equation
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| Format: | Preprint |
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2025
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| author | Ferreira, Lucas C. F. Guimarães, Ricardo M. M. |
| author_facet | Ferreira, Lucas C. F. Guimarães, Ricardo M. M. |
| contents | In this work, we investigate the blow-up of solutions to the generalized surface quasi-geostrophic (gSQG) equation in $\mathbb{R}^{2}$, within the more singular range $β\in(1,2)$ for the coupling of the velocity field. This behavior is studied under a hyperbolic setting based on the framework originally introduced by Córdoba (1998, Annals of Math. 148, 1135--52) for the classical SQG equation. Assuming that the level sets of the solution contains a hyperbolic saddle, and under suitable conditions on the solution at the origin, we obtain the existence of a time $T^{\ast}\in\mathbb{R}^{+}\cup\{\infty\}$ at which the opening angle of the saddle collapses. Moreover, we derive a lower bound for the blow-up time $T^\ast$. This geometric degeneration leads to the blow-up of the Hölder norm $\Vertθ(t)\Vert_{C^σ}$ as $t\rightarrow T^{\ast}$, for $σ\in(0, β-1)$, showing the formation of singularity in the Hölder space at time $T^{\ast}$. To the best of our knowledge, these are the first results in the literature to rigorously prove the formation of a singularity, whether in finite or infinite time, for a class of smooth solutions to the gSQG equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of hyperbolic blow-up to the generalized quasi-geostrophic equation Ferreira, Lucas C. F. Guimarães, Ricardo M. M. Analysis of PDEs 35Q35, 76U60, 35B44, 35B40 In this work, we investigate the blow-up of solutions to the generalized surface quasi-geostrophic (gSQG) equation in $\mathbb{R}^{2}$, within the more singular range $β\in(1,2)$ for the coupling of the velocity field. This behavior is studied under a hyperbolic setting based on the framework originally introduced by Córdoba (1998, Annals of Math. 148, 1135--52) for the classical SQG equation. Assuming that the level sets of the solution contains a hyperbolic saddle, and under suitable conditions on the solution at the origin, we obtain the existence of a time $T^{\ast}\in\mathbb{R}^{+}\cup\{\infty\}$ at which the opening angle of the saddle collapses. Moreover, we derive a lower bound for the blow-up time $T^\ast$. This geometric degeneration leads to the blow-up of the Hölder norm $\Vertθ(t)\Vert_{C^σ}$ as $t\rightarrow T^{\ast}$, for $σ\in(0, β-1)$, showing the formation of singularity in the Hölder space at time $T^{\ast}$. To the best of our knowledge, these are the first results in the literature to rigorously prove the formation of a singularity, whether in finite or infinite time, for a class of smooth solutions to the gSQG equation. |
| title | Existence of hyperbolic blow-up to the generalized quasi-geostrophic equation |
| topic | Analysis of PDEs 35Q35, 76U60, 35B44, 35B40 |
| url | https://arxiv.org/abs/2508.15708 |