  We prove an existence result for nonlinear diffusion equations in the
presence of a nonlocal density-dependent drift which is not necessarily
potential. The proof is constructive and based on the Helmholtz decomposition
of the drift and a splitting scheme. The splitting scheme combines transport
steps by the divergence-free part of the drift and semi-implicit minimization
steps \`a la Jordan-Kinderlherer-Otto to deal with the potential part.
