Global weak solutions to a fractional Cahn–Hilliard cross-diffusion system in lymphangiogenesis

A spectral-fractional Cahn–Hilliard cross-diffusion system, which describes the pre-patterning of lymphatic vessel morphology in collagen gels, is studied. The model consists of two higher-order quasilinear parabolic equations and describes the evolution of the fiber phase volume fraction and the solute concentration. The free energy consists of the nonconvex Flory–Huggins energy and a fractional gradient energy, modeling nonlocal long-range correlations. The existence of global weak solutions to this system in a bounded domain with Neumann boundary conditions is shown. The proof is based on a three-level approximation scheme, spectral-fractional calculus, and a priori estimates coming from the energy inequality.