Asymptotic stability of harmonic maps under the Schrödinger flow

Abstract

For Schrödinger maps from R 2 × 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 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.

Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on $${\mathbb R^2}$$

“…Even for the higher degrees m ≥ 4, this result is stronger than the previous ones [11,9,8], where the convergence was given only in time average.…”

“…Indeed, this is the essential reason for the restriction m ≥ 4 in the previous works [10,11,9]. We emphasize that the above difficulty is common for the dissipative and dispersive cases, since they share the same scaling property.…”

Traveling vortex helices for Schrödinger map equations

“…Despite some serious efforts (see e.g. [15], [16], [43], [44], [21], [9,24,39], [22,23], [19,20,17,18]) and the references therein), some basic mathematical issues such as local and global well-posedness and global in time asymptotics for the equation (1.1) remain unknown. If one is interested in one-dimensional wave (plane-wave) solutions of (1.1), that is, m : R × R → S 2 (or S 1 ), a lot were known as (1.1) becomes basically an integrable system (see [44] and [2]).…”

“…We should also note that in the case of the initial date of (1.1) contains only one vortex (one magnetic half-bubble), with its structure as described in the work of [25]- [26] very precisely, the above discussions imply that the vortex will simply stay at its center of mass, and a meaningful mathematical issue to examine would be its global stability. It is, however, unknown to authors that whether such stability result is true or not, see [22], [23], [24] for relevant discussions. On the other hand, it is relatively easy to generalize the work of [35] to the equation (1.1) for the planar ferromagnets.…”

Untitled

“…Asymptotic stability of harmonic maps of topological degree |m| ≥ 4 under the Schrödinger flow is established in [Gustafson et al 2008]. The result is extended to maps of degree |m| ≥ 3 in [Gustafson et al 2010].…”