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

TL;DR

This paper establishes the local well-posedness of the 3D Zakharov-Kuznetsov equation in certain Sobolev and Besov spaces, using sharp maximal function estimates in time-weighted spaces.

Contribution

It proves well-posedness results for the 3D Zakharov-Kuznetsov equation in new functional settings, extending previous understanding of its mathematical properties.

Findings

Well-posedness in Sobolev spaces $H^s( ^3)$ for $s>1$

Well-posedness in Besov space $B^{1,1}_2( ^3)$

Use of sharp maximal function estimates in the proof

Abstract

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation $\partial_tu+\Delta\partial_xu+ u\partial_xu=0$ in the Sobolev spaces $H^s(\R^3)$, $s>1$, as well as in the Besov space $B^{1,1}_2(\R^3)$. The proof is based on a sharp maximal function estimate in time-weighted spaces.