Global Questions for Map Evolution Equations

Just as the harmonic map equation is a geometric analogue of the classical Laplace equation for harmonic functions, so the classical linear evolution PDEs, the heat, wave, and Schrödinger equations, have geometric “map” analogues: the harmonic map heat-flow, wave map, and Schrödinger map equations. These equations are nonlinear when the target space geometry is nontrivial. Quite remarkably, these equations are all of physical (as well as mathematical) interest, at least when the target space is a 2-sphere, arising variously in the study of ferromagnets (and anti-ferromagnets), liquid crystals, and general relativity. In this article we review some recent results for map evolution equations (focusing on the Landau –Lifshitz family of equations, which includes as special cases the heat-flow and Schrödinger map) concerning the basic global questions: singularity formation vs. global regularity, and long-time asymptotics.