  The Brenier optimal map and the Knothe-Rosenblatt rearrangement are two
instances of a transport map, that is to say a map sending one measure onto
another. The main interest of the former is that it solves the
Monge-Kantorovich optimal transport problem, while the latter is very easy to
compute, being given by an explicit formula. A few years ago, Carlier,
Galichon, and Santambrogio showed that the Knothe rearrangement could be seen
as the limit of the Brenier map when the quadratic cost degenerates. In this
paper, we prove that on the torus (to avoid boundary issues), when all the data
are smooth, the evolution is also smooth, and is entirely determined by a PDE
for the Kantorovich potential (which determines the map), with a subtle initial
condition. The proof requires the use of the Nash-Moser inverse function
theorem. This result generalizes the ode discovered by Carlier, Galichon, and
Santambrogio when one measure is uniform and the other is discrete, and could
pave to way to new numerical methods for optimal transportation.
