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

TL;DR

This paper constructs a minimal mass blow-up solution for the 2D cubic Zakharov--Kuznetsov equation at low regularity, demonstrating finite time blow-up at the mass threshold, inspired by recent results in related equations.

Contribution

It provides the first construction of a minimal mass blow-up solution for the 2D cubic Zakharov--Kuznetsov equation at low regularity, extending understanding of blow-up phenomena.

Findings

Finite time blow-up at the mass threshold is possible.

Construction of minimal mass blow-up solutions at low regularity.

Extension of blow-up analysis to the 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 \phi+\partial_{x_1}(\Delta \phi+\phi^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 $\|\phi\|_{L^{2}}<\|Q\|_{L^{2}}$ are global in time. For such low regularity solutions, we study the dynamics at the threshold $\|\phi\|_{L^{2}}=\|Q\|_{L^{2}}$ 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.