Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations

We consider a nonlinear degenerate convection-diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kruzkov are obtained as the - a posteriori unique - limit points of the JKO variational approximation scheme for an associated gradient flow in the $L^2$-Wasserstein space. The equation lacks the necessary convexity properties which would allow to deduce well-posedness of the initial value problem by the abstract theory of metric gradient flows. Instead, we prove the entropy inequality directly by variational methods and conclude uniqueness by doubling of the variables.