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We establish the global existence of weak solutions for a two-speciescross-diffusion system, set on the 1-dimensional flat torus, in which theevolution of each species is governed by two mechanisms. The first of these isa diffusion which acts only on the sum of the species with a logarithmicpressure law, and the second of these is a drift term, which can differ betweenthe two species. Our main results hold under a total mixing assumption on theinitial data. This assumption, which allows the presence of vacuum, requiresspecific regularity properties for the ratio of the initial densities of thetwo species. Moreover, these regularity properties are shown to be propagatedover time. In proving the main existence result, we also establish the spatialBV regularity of solutions. In addition, our main results naturally extend tosimilar systems involving reaction terms.

On a Cross-Diffusion System with Independent Drifts and no Self-Diffusion: The Existence of Totally Mixed Solutions