Scattering of arbitrary small energy data for the three-dimensional Zakharov system
Consider the Zakharov system on $\mathbb{R}^3$, with unit ion sound speed, $$i \partial_t u - \Delta u = n u, \qquad \partial_{tt} n - \Delta n = -\Delta |u|^2,$$ for $u$ complex and $n$ real, with data $(u_0, n_0, n_1) = (u, n, \partial_t n)|_{t=0}$ in the energy space $H^1 \times L^2 \times \dot H^{-1}$, where $\|g\|_{\dot H^{-1}} = \||D|^{-1} g\|_{L^2}$. Bourgain and Colliander proved global well-posedness for data small in this space. Guo and Nakanishi proved that small radial data scatter; Guo, Lee, Nakanishi and Wang replaced the radial symmetry by angular regularity of degree one; Hani, Pusateri and Shatah proved global existence and scattering for small, spatially localized data. In four dimensions Bejenaru, Guo, Herr and Nakanishi proved scattering for small energy-space data without symmetry.
Problem. Is there $\varepsilon > 0$ such that every datum $(u_0, n_0, n_1) \in H^1(\mathbb{R}^3; \mathbb{C}) \times L^2(\mathbb{R}^3; \mathbb{R}) \times \dot H^{-1}(\mathbb{R}^3; \mathbb{R})$ with $$\|u_0\|_{H^1} + \|n_0\|_{L^2} + \|n_1\|_{\dot H^{-1}} < \varepsilon$$ produces a global solution that scatters as $t \to \pm\infty$, in the sense that for each sign there are a free Schrödinger solution $u_\pm^{\mathrm{lin}}$ and a free wave $n_\pm^{\mathrm{lin}}$ with $$\|u - u_\pm^{\mathrm{lin}}\|_{H^1} + \|n - n_\pm^{\mathrm{lin}}\|_{L^2} + \|\partial_t n - \partial_t n_\pm^{\mathrm{lin}}\|_{\dot H^{-1}} \to 0 \qquad (t \to \pm\infty)?$$ No radial symmetry, spatial weights or additional angular regularity may be imposed.
Source: Guo and Nakanishi (2026), §1, who single out energy-space scattering for nonradial data in three dimensions, even for small data, as a challenging problem; after their radial theorem (2014). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, dispersive equations, scattering, Zakharov system
References
@misc{guo2026,
author = {Guo, Zihua and Nakanishi, Kenji},
title = {Global dynamics above the ground state energy for the {3D} {Z}akharov system},
year = {2026},
eprint = {2608.14993},
archivePrefix = {arXiv},
}
@article{guo2014,
author = {Guo, Zihua and Nakanishi, Kenji},
title = {Small energy scattering for the {Z}akharov system with radial symmetry},
journal = {Int. Math. Res. Not. IMRN},
volume = {2014},
number = {9},
year = {2014},
pages = {2327--2342},
doi = {10.1093/imrn/rns296},
eprint = {1203.3959},
archivePrefix = {arXiv},
}
@article{bourgain1996,
author = {Bourgain, J. and Colliander, J.},
title = {On wellposedness of the {Z}akharov system},
journal = {Int. Math. Res. Not. IMRN},
volume = {1996},
number = {11},
year = {1996},
pages = {515--546},
doi = {10.1155/S1073792896000359},
}
@article{guo2014angular,
author = {Guo, Zihua and Lee, Sanghyuk and Nakanishi, Kenji and Wang, Chengbo},
title = {Generalized {S}trichartz estimates and scattering for {3D} {Z}akharov system},
journal = {Comm. Math. Phys.},
volume = {331},
number = {1},
year = {2014},
pages = {239--259},
doi = {10.1007/s00220-014-2006-0},
eprint = {1305.2990},
archivePrefix = {arXiv},
}
@article{hani2013,
author = {Hani, Zaher and Pusateri, Fabio and Shatah, Jalal},
title = {Scattering for the {Z}akharov system in 3 dimensions},
journal = {Comm. Math. Phys.},
volume = {322},
number = {3},
year = {2013},
pages = {731--753},
doi = {10.1007/s00220-013-1738-6},
eprint = {1206.3473},
archivePrefix = {arXiv},
}
@article{bejenaru2015,
author = {Bejenaru, Ioan and Guo, Zihua and Herr, Sebastian and Nakanishi, Kenji},
title = {Well-posedness and scattering for the {Z}akharov system in four dimensions},
journal = {Anal. PDE},
volume = {8},
number = {8},
year = {2015},
pages = {2029--2055},
doi = {10.2140/apde.2015.8.2029},
eprint = {1504.01073},
archivePrefix = {arXiv},
}
@misc{aim-zakharov-small-energy-scattering,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Scattering of arbitrary small energy data for the three-dimensional Zakharov system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/357-zakharov-small-energy-scattering.md},
note = {Entry 357 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00656, "Scattering of arbitrary small energy data for the three-dimensional Zakharov system", clopen.solutions.