Existence of global weak solutions to a Cahn-Hilliard cross-diffusion system in lymphangiogenesis

The global-in-time existence of weak solutions to a degenerate Cahn-Hilliardcross-diffusion system with singular potential in a bounded domain with no-fluxboundary conditions is proved. The model consists of two coupled parabolicfourth-order partial differential equations and describes the evolution of thefiber phase volume fraction and the solute concentration, modeling thepre-patterning of lymphatic vessel morphology. The fiber phase fractionsatisfies the segregation property if this holds initially. The existence proofis based on a three-level approximation scheme and a priori estimates comingfrom the energy and entropy inequalities. While the free energy isnonincreasing in time, the entropy is only bounded because of thecross-diffusion coupling.