Asymptotic stability of harmonic maps under the Schrödinger flow
| dc.creator | Gustafson, Stephen | |
| dc.creator | Kang, Kyungkeun | |
| dc.creator | Tsai, Tai-Peng | |
| dc.date | 2006-09-21 | |
| dc.date.accessioned | 2026-07-07T07:25:03Z | |
| dc.date.available | 2026-07-07T07:25:03Z | |
| dc.description | For Schrödinger maps from $\R^2\times\R^+$ to the 2-sphere $§^2$, it is not known if finite energy solutions can form singularities (``blowup'') in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does {\it not} occur, and global solutions converge (in a dispersive sense -- i.e. scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the ``generalized Hasimoto transform", and Strichartz (dispersive) estimates for a certain two space-dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length-scale of a nearby harmonic map. | |
| dc.description | 41 pages | |
| dc.identifier | https://arxiv.org/abs/math/0609591 | |
| dc.identifier | http://arxiv.org/abs/math/0609591 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/116599 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35Q55, 35B40 | |
| dc.title | Asymptotic stability of harmonic maps under the Schrödinger flow | |
| dc.type | text |