On the minimal Blow-up rate for the 2D generalized Zakharov- Kuznetsov model
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| Format: | Preprint |
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2025
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| _version_ | 1866911570438651904 |
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| author | Trespalacios, Jessica |
| author_facet | Trespalacios, Jessica |
| contents | In this note we consider the generalized Zakharov-Kuznetsov equation in $\mathbb R^2$, for initial conditions in the Sobolev space $H^s$ with $s>3/4.$ Assuming that there is a blow-up solution at finite time $T^{*}$, we obtain a lower bound for the blow-up rate of that solution, expressed in terms of a lower bound for the $H^s$ norm of the solution. In the particular case of the modified Zakharov-Kuznetsov equation, {\color{teal} a nontrivial gap is found between conjectured blow-up rates and our results.} The analysis is based on properly quantifying the linear estimates given by Faminskii \cite{Faminskii}, as well as the local well-posedness theory of Linares and Pastor \cite{Linares2009,LinaresPastor}, combined with an argument developed by Weissler \cite{Weissler} and {\color{teal} Colliander, Czuback and Sulem} \cite{Colliander} in the context of the semilinear heat equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the minimal Blow-up rate for the 2D generalized Zakharov- Kuznetsov model Trespalacios, Jessica Analysis of PDEs In this note we consider the generalized Zakharov-Kuznetsov equation in $\mathbb R^2$, for initial conditions in the Sobolev space $H^s$ with $s>3/4.$ Assuming that there is a blow-up solution at finite time $T^{*}$, we obtain a lower bound for the blow-up rate of that solution, expressed in terms of a lower bound for the $H^s$ norm of the solution. In the particular case of the modified Zakharov-Kuznetsov equation, {\color{teal} a nontrivial gap is found between conjectured blow-up rates and our results.} The analysis is based on properly quantifying the linear estimates given by Faminskii \cite{Faminskii}, as well as the local well-posedness theory of Linares and Pastor \cite{Linares2009,LinaresPastor}, combined with an argument developed by Weissler \cite{Weissler} and {\color{teal} Colliander, Czuback and Sulem} \cite{Colliander} in the context of the semilinear heat equations. |
| title | On the minimal Blow-up rate for the 2D generalized Zakharov- Kuznetsov model |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.09157 |