Cross-diffusion systems model species whose diffusion depends on the aggregate density: each species spreads according to how many total particles are present, not just its own concentration. When the species also have independent drifts — each pushed by a different potential — the analysis becomes significantly harder because the drifts can separate species that diffusion tries to mix.
Previous existence results required total mixing: the initial data had to have overlapping supports, ensuring the species were already intermingled. This paper removes that assumption entirely. For logarithmic and fast-diffusion pressure laws in one dimension with no-flux boundaries, global existence of solutions is established for general initial data with finite bounded variation, with no structural constraints on their supports.
The species can start completely separated — non-overlapping intervals — and the cross-diffusion system still has a global solution. The drifts push them in different directions while the shared pressure mediates their interaction. The result gives a full resolution of problems left open by recent work: bounded variation is enough, total mixing is unnecessary. The diffusion creates mixing over time without requiring it at the start.