Well-posedness results for the 3D Zakharov-Kuznetsov equation

Abstract

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation ${\partial }_{t}u+\Delta {\partial }_{x}u+u{\partial }_{x}u=0$ in the Sobolev spaces $H^s({\mathbb{R}}^3)$, $s>1$, as well as in the Besov space $B_2^{1,1}({\mathbb{R}}^3)$. The proof is based on a sharp maximal function estimate in time-weighted spaces.

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