Equivariant heat and Schrodinger flows from Euclidean space to complex projective space

We study the equivariant harmonic map heat flow, Schrodinger maps equation, and gen-eralized Landau-Lifshitz equation from Cn to CPn. By means of a careful geometric analysis, we determine a new, highly useful representation of the problem in terms of a PDE for radial functions from Cn to S2. Using this new representation, we are able to write explicit formulas for the har-monic maps in this context, and prove that they all have infinite energy. We show that the PDEs admit a family of self-similar solutions with smooth profiles; these solutions again have infinite energy, and give an example of regularity breakdown. Then, using a variant of the Hasimoto transforma-tion applied to our new equation for the dynamics, we prove a small-data global well-posedness result when n = 2. This is, to the best of our knowledge, the first global well-posedness result for Schrodinger maps when the complex dimension of the target is greater than 1. In the final section we study a special case of the harmonic map heat flow corresponding to initial data valued in one great circle. We show that the n = 2 case of this problem is a borderline case for the standard classification theory for PDEs of its type.