A cross-diffusion system obtained via (convex) relaxation in the JKO scheme

In this paper, we start from a very natural system of cross-diffusion equations, which can be seen as gradient flow for the Wasserstein distance certain functional. Unfortunately, is not well-posed, consequence fact that underlying functional lower semi-continuous. We then consider relaxation functional, and prove existence solution in suitable sense (the relaxed functional). This has also structure, but mixture between two different regimes, are determined by relaxation, makes study non-trivial.