Passage to the limit for general convex energies in the JKO scheme

Establish a rigorous passage to the limit that yields a weak formulation for Wasserstein gradient flows of cross-diffusion systems when the energy has the general form E(u)=∫Ω E(u) dx with E: ℝ^N → [0,∞) a smooth convex function, by proving the necessary chain-rule/variational derivative justifications in the Jordan–Kinderlehrer–Otto scheme without assuming the McCann condition or a quadratic structure.

Background

The existence proof in Section 3.2 relies on the Jordan–Kinderlehrer–Otto (JKO) scheme for the quadratic energy corresponding to the Busenberg–Travis system. A key step is justifying the chain rule and passing to the limit in the energy variation under perturbations induced by push-forward maps, which enables deriving the weak formulation from the discrete Euler–Lagrange conditions.

For more general energies E(u)=∫Ω E(u) dx with E a smooth convex function on ℝ^N, this justification is only known in special cases: when E satisfies the McCann displacement convexity condition (allowing monotone convergence) or when E has a quadratic structure (allowing weak–strong arguments). The authors note that, beyond these cases, carrying out the limit to obtain a weak formulation is delicate and remains unresolved.