One cubic NLS trajectory with unbounded Sobolev norm
On $\mathbb{T}^2 = (\mathbb{R}/2\pi\mathbb{Z})^2$ consider the defocusing cubic nonlinear Schrödinger equation $$i\partial_t u + \Delta u = |u|^2 u, \qquad u(0) = u_0,$$ and write $\|u\|_{H^2}^2 = \sum_{k \in \mathbb{Z}^2} (1 + |k|^2)^2 |\widehat{u}(k)|^2$. Bourgain's local well-posedness theory and the conservation of mass and energy give a global smooth solution for smooth data, with $H^1$ norm bounded for all time (1993), and Bourgain proved polynomial-in-time upper bounds for its higher Sobolev norms (1996). Colliander, Keel, Staffilani, Takaoka and Tao proved that for every $s > 1$, $K \gg 1$ and $0 < \delta \ll 1$ there is a global smooth solution with $\|u(0)\|_{H^s} \le \delta$ and $\|u(T)\|_{H^s} \ge K$ at some time $T > 0$ (2010). On $\mathbb{R} \times \mathbb{T}^d$ with $d \ge 2$, Hani, Pausader, Tzvetkov and Visciglia constructed global solutions of the cubic equation whose high Sobolev norms grow without bound (2015).
Problem. Is there $u_0 \in C^\infty(\mathbb{T}^2; \mathbb{C})$ whose global solution satisfies $$\limsup_{t \to \infty} \|u(t)\|_{H^2(\mathbb{T}^2)} = \infty?$$ A single initial datum and its forward trajectory must realise the unbounded growth.
Source: Bourgain (2000), for the H^s norms with s > 1; see Colliander, Keel, Staffilani, Takaoka and Tao (2010). Stated for smooth data and the H^2 norm by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, energy cascades, nonlinear Schrödinger equations
References
@incollection{bourgain2000,
author = {Bourgain, J.},
title = {Problems in {H}amiltonian {PDE}'s},
booktitle = {Visions in Mathematics: {GAFA} 2000 Special Volume, Part {I}},
editor = {Alon, N. and Bourgain, J. and Connes, A. and Gromov, M. and Milman, V.},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {2000},
pages = {32--56},
doi = {10.1007/978-3-0346-0422-2_2},
note = {Geom. Funct. Anal., Special Volume 2000; the DOI is that of the 2010 reprint},
}
@article{colliander2010,
author = {Colliander, J. and Keel, M. and Staffilani, G. and Takaoka, H. and Tao, T.},
title = {Transfer of energy to high frequencies in the cubic defocusing nonlinear {S}chr{\"o}dinger equation},
journal = {Invent. Math.},
volume = {181},
number = {1},
year = {2010},
pages = {39--113},
doi = {10.1007/s00222-010-0242-2},
eprint = {0808.1742},
archivePrefix = {arXiv},
}
@article{bourgain1993,
author = {Bourgain, J.},
title = {Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. {I}. {S}chr{\"o}dinger equations},
journal = {Geom. Funct. Anal.},
volume = {3},
number = {2},
year = {1993},
pages = {107--156},
doi = {10.1007/BF01896020},
}
@article{bourgain1996,
author = {Bourgain, J.},
title = {On the growth in time of higher {S}obolev norms of smooth solutions of {H}amiltonian {PDE}},
journal = {Int. Math. Res. Not.},
volume = {1996},
number = {6},
year = {1996},
pages = {277--304},
doi = {10.1155/S1073792896000207},
}
@article{hani2015,
author = {Hani, Zaher and Pausader, Benoit and Tzvetkov, Nikolay and Visciglia, Nicola},
title = {Modified scattering for the cubic {S}chr{\"o}dinger equation on product spaces and applications},
journal = {Forum Math. Pi},
volume = {3},
year = {2015},
pages = {e4},
doi = {10.1017/fmp.2015.5},
eprint = {1311.2275},
archivePrefix = {arXiv},
}
@misc{aim-cubic-nls-unbounded-orbit,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{One cubic NLS trajectory with unbounded Sobolev norm}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/226-cubic-nls-unbounded-orbit.md},
note = {Entry 226 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-00585, "One cubic NLS trajectory with unbounded Sobolev norm", clopen.solutions.