Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation

Abstract

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 $\Vert \varphi {\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}$ are global in time. For such low regularity solutions, we study the dynamics at the threshold $\Vert \varphi {\Vert }_{L^2}=\Vert Q{\Vert }_{L^2}$ 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.

56 pages