Global existence for a system without self-diffusion and different mobilities

We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a linear pressure law and different mobilities. For arbitrary bounded non-negative initial data, we show that any good approximation scheme, yields existence of global weak solutions. More precisely, we introduce a notion of $\mathit{\text{admissible}}\mathit{\text{approximation}}\mathit{\text{sequence}}$ and show that any such sequence admits a subsequence converging to a weak solution of the system. The strategy relies on entropy estimates and the div–curl lemma, in the framework of Young measures.