Scattering of the 3D Zakharov-Kuznetsov equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911621898567680 |
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| author | Anjolras, Philippe |
| author_facet | Anjolras, Philippe |
| contents | We consider the Zakharov-Kuznetsov equation in space dimension 3: \[ \left\{ \begin{array}{l} \partial_t u + \partial_x Δu + \partial_x \frac{u^2}{2} = 0 \\ u(t = 0) = u_0 \end{array} \right. \] where $u : (t, x, y) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R}^2 \mapsto u(t, x, y) \in \mathbb{R}$, and $Δ= \partial_x^2 + Δ_y$ is the full Laplacian. We show that, for any $u_0$ satisfying \[ \Vert (1 + x^2 + |y|^2) u_0 \Vert_{H^1} \ll 1 \] then the global solution exhibits scattering in $H^1$. This is done using the method of space-time resonances, and more precisely the partial symmetries approach [GPW23] in order to treat the anisotropy. We introduce well suited anisotropic weighted norms, prove dispersive decay estimates adapted to these norms and an a priori estimate allowing to close by a bootstrap argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22957 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scattering of the 3D Zakharov-Kuznetsov equation Anjolras, Philippe Analysis of PDEs We consider the Zakharov-Kuznetsov equation in space dimension 3: \[ \left\{ \begin{array}{l} \partial_t u + \partial_x Δu + \partial_x \frac{u^2}{2} = 0 \\ u(t = 0) = u_0 \end{array} \right. \] where $u : (t, x, y) \in \mathbb{R} \times \mathbb{R} \times \mathbb{R}^2 \mapsto u(t, x, y) \in \mathbb{R}$, and $Δ= \partial_x^2 + Δ_y$ is the full Laplacian. We show that, for any $u_0$ satisfying \[ \Vert (1 + x^2 + |y|^2) u_0 \Vert_{H^1} \ll 1 \] then the global solution exhibits scattering in $H^1$. This is done using the method of space-time resonances, and more precisely the partial symmetries approach [GPW23] in order to treat the anisotropy. We introduce well suited anisotropic weighted norms, prove dispersive decay estimates adapted to these norms and an a priori estimate allowing to close by a bootstrap argument. |
| title | Scattering of the 3D Zakharov-Kuznetsov equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.22957 |