Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

TL;DR

This paper studies the blow-up behavior of solutions to the 2D $L^2$ critical Zakharov-Kuznetsov equation, classifying solutions into stable, blow-up, or diverging types, using methods inspired by other critical PDEs.

Contribution

It extends blow-up analysis techniques to the 2D Zakharov-Kuznetsov equation, providing a classification of solution behaviors near the critical mass.

Findings

Identifies three possible solution behaviors: stability, blow-up, divergence.

Constructs explicit blow-up solutions via solitary wave bubbling.

Demonstrates universal blow-up behavior and stability of solutions.

Abstract

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} (\Delta u+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.