Blow-up in finite or infinite time of the 2D cubic Zakharov-Kuznetsov equation

Abstract

We prove that near-threshold negative energy solutions to the 2D cubic ($L^2$-critical) focusing Zakharov-Kuznetsov (ZK) equation blow-up in finite or infinite time. The proof consists of several steps. First, we show that if the blow-up conclusion is false, there are negative energy solutions arbitrarily close to the threshold that are globally bounded in $H^1$ and are spatially localized, uniformly in time. In the second step, we show that such solutions must in fact be exact remodulations of the ground state, and hence, have zero energy, which is a contradiction. This second step, a nonlinear Liouville theorem, is proved by contradiction, with a limiting argument producing a nontrivial solution to a (linear) linearized ZK equation obeying uniform-in-time spatial localization. Such nontrivial linear solutions are excluded by a local-viral space-time estimate. The general framework of the argument is modeled on Merle [29] and Martel & Merle [24], who treated the 1D problem of the $L^2$-critical gKdV equation. Several new features are introduced here to handle the 2D ZK case.