Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| author | Bozgan, Francisc Ghoul, Tej-Eddine Masmoudi, Nader Yang, Kai |
| author_facet | Bozgan, Francisc Ghoul, Tej-Eddine Masmoudi, Nader Yang, Kai |
| contents | We investigate the blow-up dynamics for the $L^2$ critical two-dimensional Zakharov-Kuznetsov equation \begin{equation*} \begin{cases} \partial_t u+\partial_{x_1} (Δu+u^3)=0, \mbox{ } x=(x_1,x_2)\in \mathbb{R}^2, \mbox{ } t \in \mathbb{R}\\ u(0,x_1,x_2)=u_0(x_1,x_2)\in H^1(\mathbb{R}^2), \end{cases} \end{equation*} with initial data $u_0$ slightly exceeding the mass of the ground state $Q$. Employing methodologies analogous to the Martel-Merle-Raphael blow-up theory for $L^2$ critical equations, more precisely for the critical NLS equation and the quintic generalized Korteweg-de Vries equation, we categorize the solution behaviors into three outcomes: asymptotic stability, finite-time blow-up, or divergence from the soliton's vicinity. The construction of the blow-up solution involves the bubbling of the solitary wave which ensures the universal behavior and stability of the blow-up. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_06568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation Bozgan, Francisc Ghoul, Tej-Eddine Masmoudi, Nader Yang, Kai Analysis of PDEs 35Q53, 35B44, 35B33, 35C08 We investigate the blow-up dynamics for the $L^2$ critical two-dimensional Zakharov-Kuznetsov equation \begin{equation*} \begin{cases} \partial_t u+\partial_{x_1} (Δu+u^3)=0, \mbox{ } x=(x_1,x_2)\in \mathbb{R}^2, \mbox{ } t \in \mathbb{R}\\ u(0,x_1,x_2)=u_0(x_1,x_2)\in H^1(\mathbb{R}^2), \end{cases} \end{equation*} with initial data $u_0$ slightly exceeding the mass of the ground state $Q$. Employing methodologies analogous to the Martel-Merle-Raphael blow-up theory for $L^2$ critical equations, more precisely for the critical NLS equation and the quintic generalized Korteweg-de Vries equation, we categorize the solution behaviors into three outcomes: asymptotic stability, finite-time blow-up, or divergence from the soliton's vicinity. The construction of the blow-up solution involves the bubbling of the solitary wave which ensures the universal behavior and stability of the blow-up. |
| title | Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation |
| topic | Analysis of PDEs 35Q53, 35B44, 35B33, 35C08 |
| url | https://arxiv.org/abs/2406.06568 |