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

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 $k\ge 2$. In particular, we show that the modified Zakharov-Kuznetsov equation is locally well-posed in $H^s(\mathbb{R}\times \mathbb{T})$ for all $s>\frac{11}{24}$.