Low-regularity global well-posedness theory for the generalized Zakharov-Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$ and polynomial growth of higher Sobolev norms

Abstract

We address the Cauchy problem for the $k$-generalized Zakharov-Kuznetsov equation ($k$-gZK) posed on $\mathbb{R}^2$ and on $\mathbb{R} \times \mathbb{T}$. By applying established and recently developed linear and bilinear Strichartz-type estimates within the framework of the $I$-method, we obtain the following results: $\bullet$ The Zakharov-Kuznetsov equation is globally well-posed in $H^s(\mathbb{R} \times \mathbb{T})$ for every $s>\frac{11}{13}$. $\bullet$ The modified Zakharov-Kuznetsov equation is globally well-posed in $H^s(\mathbb{R}^2)$ for every $s>\frac{2}{3}$ and in $H^s(\mathbb{R} \times \mathbb{T})$ for every $s>\frac{36}{49}$. Moreover, we show that the $H^s(\mathbb{R} \times \mathbb{T})$-norm of smooth global real-valued solutions of $k$-gZK grows at most polynomially in time for every $k\geq 1$.