The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces

TL;DR

This paper establishes local well-posedness for the Benjamin-Ono-Zakharov-Kuznetsov equation in low regularity Sobolev spaces, extending understanding of its initial-value problem in two dimensions.

Contribution

It proves local well-posedness in $L^2$-based Sobolev spaces for the first time, using methods adapted from the Benjamin-Ono equation.

Findings

Well-posedness in $H^s( ^2)$ for $s>11/8$

Extension to some weighted Sobolev spaces

Methodology based on Benjamin-Ono equation techniques

Abstract

In this paper we study the initial-value problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation. Such equation appears as a two-dimensional generalization of the Benjamin-Ono equation when transverse effects are included via weak dispersion of Zakharov-Kuznetsov type. We prove that the initial-value problem is locally well-posed in the usual $L^2(\R^2)$-based Sobolev spaces $H^{s}(\R^2)$, $s>11/8$, and in some weighted Sobolev spaces. To obtain our results, most of the arguments are accomplished taking into account the ones for the Benjamin-Ono equation.