Remarks on continuity equations with nonlinear diffusion and nonlocal drifts

This paper is devoted to existence and uniqueness results for some classes of nonlinear diffusion equations in the presence of a regular   drift term. These equations may be viewed as regular perturbations of Wasserstein gradient flows but the drift terms are not necessarily gradients (which makes it difficult to use Wasserstein gradient flows techniques). We obtain existence by a regularization procedure and parabolic energy estimates and address the uniqueness issue by an elementary H−1H−1 contraction argument if the diffusion is nondegenerate. Our arguments directly extend to systems with diagonal nonlinear diffusions which are coupled through regular drifts.