  We prove the existence of weak solutions to a system of two diffusion
equations that are coupled by a pointwise volume constraint. The time evolution
is given by gradient dynamics for a free energy functional. Our primary example
is a model for the demixing of polymers, the corresponding energy is the one of
Flory, Huggins and deGennes. Due to the non-locality in the equations, the
dynamics considered here is qualitatively different from the one found in the
formally related Cahn-Hilliard equations.
  Our angle of attack is from the theory of optimal mass transport, that is, we
consider the evolution equations for the two components as two gradient flows
in the Wasserstein distance with one joint energy functional that has the
volume constraint built in. The main difference to our previous work
arXiv:1712.06446 is the nonlinearity of the energy density in the gradient
part, which becomes singular at the interface between pure and mixed phases.
