A cross-diffusion system with independent drifts and fast diffusion

We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a fast-diffusion law with exponent $0<\alpha \le 1$ and different external potentials. For arbitrary non-negative $L^1$ initial data with bounded entropy and a mixing condition we prove the existence of global weak solutions. This extends the recent result of Mészáros, Parker from the linear diffusion ($\alpha =1$) to the fast-diffusion.