| We investigate the analytic behavior of Schrodinger maps, which are solutions of a geometric partial differential equation. This equation is a natural generalization of the Landau-Lifshitz equation; therefore, we also study questions related to this physically relevant special case. Due to the geometric constraints of these PDEs, the equations are highly nonlinear, and we must repeatedly appeal to their geometric structure to overcome the apparent analytical difficulties of proving well-posedness.;We consider Schrodinger maps from Rd x [0, T] into Kahler manifolds with bounded geometry. Our idea is to approximate the Schrodinger map equation by a sequence of wave maps. Relying on known results for wave maps, we prove local existence of Schrodinger maps in Sobolev spaces of integer order l > d2 + 1. We use standard energy estimate methods; however, we differentiate covariantly in order to preserve the structure of the equation. Also taking advantage of the geometry of the problem, we prove uniqueness by using parallel transport to compare two solutions on a target manifold. Moreover, by considering initial data with spatial decay, we extend the above results to lower order Sobolev spaces.;When the target manifold is the unit sphere, the Schrodinger map equation is equivalent to the Landau-Lifshitz equation, a simple model of ferromagnetic materials. If we add the effects of an electromagnetic field and couple the Landau-Lifshitz equation to Maxwell's equations, we can show local existence for this system, also by energy methods. |
