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

TL;DR

This paper proves the existence of global weak solutions for a complex fractional Cahn-Hilliard cross-diffusion system modeling lymphangiogenesis, incorporating nonlocal interactions and higher-order quasilinear parabolic equations.

Contribution

It introduces a novel spectral-fractional approach to establish global weak solutions for a nonlocal, fractional Cahn-Hilliard system relevant to biological tissue modeling.

Findings

Existence of global weak solutions proven

Utilization of spectral-fractional calculus in analysis

Energy estimates ensure solution stability

Abstract

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 no-flux 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.