Nonexistence of minimal mass blow-up solution for the 2D cubic Zakharov-Kuznetsov equation

TL;DR

This paper proves that the 2D cubic Zakharov-Kuznetsov equation does not admit minimal mass blow-up solutions in the energy space, contrasting with similar results for the gKdV equation.

Contribution

It establishes the nonexistence of minimal mass blow-up solutions for the 2D cubic Zakharov-Kuznetsov equation using refined ODE and energy-virial methods.

Findings

No finite/infinite time minimal mass blow-up solutions exist.

The proof employs a modified energy-virial Lyapunov functional.

Contrasts with known results for the gKdV equation.

Abstract

For the 2D 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*} we prove that there exist no finite/infinite time blow-up solution with minimal mass in the energy space. This nonexistence result is in contrast to the one obtained by Martel-Merle-Rapha\"el [17] for the mass-critical generalized Korteweg-de Vries (gKdV) equation. The proof relies on a refined ODE argument related to the modulation theory and a modified energy-virial Lyapunov functional with a monotonicity property.