  The theory of (Wasserstein) gradient flows in the space of probability
measures has made enormous progress over the last twenty years. Nonetheless,
many partial differential equations (PDEs) of interest do not have gradient
flow structure and, a priori, the theory is not applicable. In this paper, we
develop a time-discrete entropic regularised variational scheme for a general
class of such non-gradient PDEs. We prove the convergence of the scheme and
illustrate the breadth of the proposed framework with concrete examples
including the non-linear kinetic Fokker-Planck (Kramers) equation and a
nonlinear degenerate diffusion of Kolmogorov type.
