Asymptotic stability of harmonic maps under the Schrödinger flow

Details Asymptotic stability of harmonic maps under the Schrödinger flow Asymptotic stability of harmonic maps under the Schrödinger flow

2006-09-21

arxiv.org/abs/math/0609591v1

Mathematics

Analysis of PDEs

41 pages

Scientific paper

For Schr\"odinger maps from $\R^2\times\R^+$ to the 2-sphere $\S^2$, 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 Schr\"odinger 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.

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