Persisting entropy structure for nonlocal cross-diffusion systems

For cross-diffusion systems possessing an entropy (i.e. a Lyapunov functional) we study nonlocal versions and exhibit sufficient conditions to ensure that the nonlocal version inherits the entropy structure. These nonlocal systems can be understood as population models per se or as approximation of the classical ones. With the preserved entropy, we can rigorously link the approximating nonlocal version to the classical local system. From a modelling perspective this gives a way to prove a derivation of the model and from a PDE perspective this provides a regularisation scheme to prove the existence of solutions. A guiding example is the SKT model [18] and in this context we answer positively to a question raised by Fontbona and Meleard in [9] and thus provide a full derivation.