Strichartz estimates for the generalized Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$ and applications

TL;DR

This paper establishes new Strichartz estimates for the generalized Zakharov-Kuznetsov equation on a mixed real-torus domain, leading to improved well-posedness results for the equation in low regularity Sobolev spaces.

Contribution

It introduces almost optimal linear and bilinear estimates for the equation, lowering the regularity threshold for local well-posedness on RT.

Findings

Well-posedness in H^s for s > 11/24

New linear L^4 estimate established

Bilinear refinements improve regularity thresholds

Abstract

In this article, we address the Cauchy problem associated with the $k$-generalized Zakharov-Kuznetsov equation posed on $\mathbb{R} \times \mathbb{T}$. By establishing an almost optimal linear $L^4$-estimate, along with a family of bilinear refinements, we significantly lower the well-posedness threshold for all $k \geq 2$. In particular, we show that the modified Zakharov-Kuznetsov equation is locally well-posed in $H^s(\mathbb{R} \times \mathbb{T})$ for all $s > \frac{11}{24}$.