A note on the well-posedness in the energy space for the generalized ZK equation posed on $\mathbb{R}\times\mathbb{T}$

TL;DR

This paper establishes local well-posedness in the energy space for the generalized Zakharov-Kuznetsov equation on a mixed domain and proves global solutions under small initial data conditions using a sharp inequality.

Contribution

It extends well-posedness results to the energy space for the equation on  imes  with any nonlinearity power, and derives global solutions with small initial data.

Findings

Proved local well-posedness in the energy space for all nonlinearities with k  2.

Established global solutions under small initial data assumptions.

Derived a sharp Gagliardo-Nirenberg inequality for the analysis.

Abstract

In this note, we prove the local well-posedness in the energy space of the $k$-generalized Zakharov-Kuznetsov equation posed on $ \R\times \T $ for any power non-linearity $ k\ge 2$. Moreover, we obtain global solutions under a precise smallness assumption on the initial data by proving a sharp Gagliardo Nirenberg type inequality.