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
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912982765666304 |
|---|---|
| author | Nowicki-Koth, Jakob |
| author_facet | Nowicki-Koth, Jakob |
| contents | 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 $k\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24737 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 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 Nowicki-Koth, Jakob Analysis of PDEs 35Q53 (Primary) 35R05 (Secondary) 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 $k\geq 1$. |
| title | 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 |
| topic | Analysis of PDEs 35Q53 (Primary) 35R05 (Secondary) |
| url | https://arxiv.org/abs/2603.24737 |