Well-posedness for the two dimensional generalized Zakharov-Kuznetsov equation in anisotropic weighted Sobolev spaces

TL;DR

This paper establishes well-posedness for the 2D generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces using contraction mapping and new commutator formulas.

Contribution

It introduces a novel approach combining contraction mapping with a new commutator formula for anisotropic fractional weights.

Findings

Well-posedness in fractional weighted Sobolev spaces

New commutator formula for anisotropic weights

Extension of previous Sobolev space results

Abstract

We consider the well-posedness of the initial value problem associated to the k-generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces. Our method of proof is based on the contraction mapping principle and it mainly relies on the well-posedness results recently obtained for this equation in the Sobolev spaces H^s(\R^2) and a new pointwise commutator type formula involving the group induced by the linear part of the equation and the fractional anisotropic weights to be considered