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arXiv · 2603.24737

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

Jakob Nowicki-Koth
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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 $k\geq 1$.

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Jakob Nowicki-Koth. 2026-03-25. 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. https://arxiv.org/abs/2603.24737

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