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We prove that near-threshold negative energy solutions to the 2D cubic($L^2$-critical) focusing Zakharov-Kuznetsov (ZK) equation blow-up in finite orinfinite time. The proof consists of several steps. First, we show that if theblow-up conclusion is false, there are negative energy solutions arbitrarilyclose to the threshold that are globally bounded in $H^1$ and are spatiallylocalized, uniformly in time. In the second step, we show that such solutionsmust in fact be exact remodulations of the ground state, and hence, have zeroenergy, which is a contradiction. This second step, a nonlinear Liouvilletheorem, is proved by contradiction, with a limiting argument producing anontrivial solution to a (linear) linearized ZK equation obeyinguniform-in-time spatial localization. Such nontrivial linear solutions areexcluded by a local-viral space-time estimate. The general framework of theargument is modeled on Merle [29] and Martel & Merle [24], who treated the 1Dproblem of the $L^2$-critical gKdV equation. Several new features areintroduced here to handle the 2D ZK case.

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