Equivariant heat and Schr\"odinger flows from Euclidean space to complex projective space

Abstract

We study the equivariant harmonic map heat flow, Schr\"odinger maps equation, and generalized Landau-Lifshitz equation from $\mathbb{C}^n$ to $\mathbb{C}\mathbb{P}^n$. 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 $\mathbb{C}^n$ to $\mathbb{S}^2$. Using this new representation, we are able to write explicit formulae for the harmonic 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 transformation applied to our new equation for the dynamics, we prove a small-data global wellposedness result when $n=2$. This is, to the best of our knowledge, the first global wellposedness result for Schr\"odinger maps when the complex dimension of the target is greater than one. 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 PDE of its type.