Global smooth Schrödinger maps below the degree-zero energy threshold
For maps $\phi : \mathbb{R}^2 \to \mathbb{S}^2$ let $$E(\phi) = \frac12 \int_{\mathbb{R}^2} |\nabla \phi|^2 \, dx, \qquad \deg \phi = \frac{1}{4\pi} \int_{\mathbb{R}^2} \phi \cdot (\partial_1 \phi \times \partial_2 \phi) \, dx,$$ and consider the Schrödinger map equation into $\mathbb{S}^2$, the isotropic Landau–Lifshitz equation without damping, $$\partial_t \phi = \phi \times \Delta \phi, \qquad \phi(0) = \phi_0.$$ The energy is conserved and invariant under the scaling $\phi(t, x) \mapsto \phi(\lambda^2 t, \lambda x)$, and nonconstant harmonic maps of finite energy have energy at least $4\pi$. Bejenaru, Ionescu, Kenig and Tataru proved global existence of smooth solutions for smooth data with $\|\phi_0 - q\|_{\dot H^1}$ small, and global existence and scattering for 1-equivariant data with $E(\phi_0) < 4\pi$. Merle, Raphaël and Rodnianski constructed smooth 1-equivariant data of degree one, arbitrarily close to a harmonic map of energy $4\pi$ in the energy norm, whose solutions blow up in finite time.
Problem. Let $q \in \mathbb{S}^2$ and let $\phi_0$ be smooth with $\phi_0 - q \in \bigcap_{k \ge 1} H^k(\mathbb{R}^2)$, $E(\phi_0) < 8\pi$ and $\deg \phi_0 = 0$. Must the solution extend uniquely to all $t \in \mathbb{R}$, with $\phi - q \in C(\mathbb{R}; H^k)$ for every integer $k \ge 1$? No rotational symmetry or a priori spacetime bound is assumed.
Source: Tataru (2012), §5, the strong threshold conjecture (Conjecture 3), which asks for global well-posedness and scattering for degree-zero data of energy below twice that of the least-energy nontrivial harmonic map; the 8π threshold for degree-zero equivariant maps also in Bejenaru, Ionescu, Kenig and Tataru (2013). Stated by AIM — Open Applied Problems as its global-regularity part.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG, MP
- Keywords
- AIM Open Applied Problems, energy-critical dispersive equations, Landau–Lifshitz equation, Schrödinger maps
References
@article{tataru2012,
author = {Tataru, Daniel},
title = {Schr{\"o}dinger maps},
journal = {Journ{\'e}es {\'E}quations aux d{\'e}riv{\'e}es partielles},
year = {2012},
note = {Expos{\'e} no. IX},
pages = {1--11},
doi = {10.5802/jedp.92},
}
@article{bejenaru2013,
author = {Bejenaru, I. and Ionescu, A. D. and Kenig, C. E. and Tataru, D.},
title = {Equivariant {S}chr{\"o}dinger maps in two spatial dimensions},
journal = {Duke Math. J.},
volume = {162},
number = {11},
year = {2013},
pages = {1967--2025},
doi = {10.1215/00127094-2293611},
eprint = {1112.6122},
archivePrefix = {arXiv},
}
@article{bejenaru2011,
author = {Bejenaru, I. and Ionescu, A. D. and Kenig, C. E. and Tataru, D.},
title = {Global {S}chr{\"o}dinger maps in dimensions $d \ge 2$: small data in the critical {S}obolev spaces},
journal = {Ann. of Math. (2)},
volume = {173},
number = {3},
year = {2011},
pages = {1443--1506},
doi = {10.4007/annals.2011.173.3.5},
eprint = {0807.0265},
archivePrefix = {arXiv},
}
@article{merle2013,
author = {Merle, Frank and Rapha{\"e}l, Pierre and Rodnianski, Igor},
title = {Blowup dynamics for smooth data equivariant solutions to the critical {S}chr{\"o}dinger map problem},
journal = {Invent. Math.},
volume = {193},
number = {2},
year = {2013},
pages = {249--365},
doi = {10.1007/s00222-012-0427-y},
eprint = {1102.4308},
archivePrefix = {arXiv},
}
@misc{aim-schrodinger-map-threshold,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global smooth Schrödinger maps below the degree-zero energy threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/355-schrodinger-map-threshold.md},
note = {Entry 355 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-00641, "Global smooth Schrödinger maps below the degree-zero energy threshold", clopen.solutions.