  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.
