Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915459887005696 |
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| author | Lan, Yang Yuan, Xu |
| author_facet | Lan, Yang Yuan, Xu |
| contents | In this article, we construct a minimal mass blow-up solution of the two-dimensional cubic (mass-critical) Zakharov--Kuznetsov equation:
\begin{equation*}
\partial_t ϕ+\partial_{x_1}(Δϕ+ϕ^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^2.
\end{equation*}
Let $s>\frac{3}{4}$. Bhattacharya-Farah-Roudenko [2] show that $H^{s}$ solutions with $\|ϕ\|_{L^{2}}<\|Q\|_{L^{2}}$ are global in time. For such low regularity solutions, we study the dynamics at the threshold $\|ϕ\|_{L^{2}}=\|Q\|_{L^{2}}$ and demonstrate that finite time blow-up singularity formation may occur. This result and its proof are inspired by the recent blow-up result [29] for the mass-critical gKdV equation. This result is also complement of previous result [6] for nonexistence of minimal mass blow-up element in the energy space $H^{1}(\mathbb{R}^{2})$ of the two-dimensional cubic Zakharov--Kuznetsov equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation Lan, Yang Yuan, Xu Analysis of PDEs In this article, we construct a minimal mass blow-up solution of the two-dimensional cubic (mass-critical) Zakharov--Kuznetsov equation: \begin{equation*} \partial_t ϕ+\partial_{x_1}(Δϕ+ϕ^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^2. \end{equation*} Let $s>\frac{3}{4}$. Bhattacharya-Farah-Roudenko [2] show that $H^{s}$ solutions with $\|ϕ\|_{L^{2}}<\|Q\|_{L^{2}}$ are global in time. For such low regularity solutions, we study the dynamics at the threshold $\|ϕ\|_{L^{2}}=\|Q\|_{L^{2}}$ and demonstrate that finite time blow-up singularity formation may occur. This result and its proof are inspired by the recent blow-up result [29] for the mass-critical gKdV equation. This result is also complement of previous result [6] for nonexistence of minimal mass blow-up element in the energy space $H^{1}(\mathbb{R}^{2})$ of the two-dimensional cubic Zakharov--Kuznetsov equation. |
| title | Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.16960 |