  We consider a nonlinear degenerate convection-diffusion equation with
inhomogeneous convection and prove that its entropy solutions in the sense of
Kru\v{z}kov 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.
