  This paper is devoted to existence and uniqueness results for classes of
nonlinear diffusion equations (or systems) which may be viewed as regular
perturbations of Wasserstein gradient flows. First, in the case. where the
drift is a gradient (in the physical space), we obtain existence by a
semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case,
we derive existence from a regularization procedure and parabolic energy
estimates. We also address the uniqueness issue by a displacement convexity
argument.
