Analysis and structure-preserving approximation of a Cahn-Hilliard-Forchheimer system with solution-dependent mass and volume source

Abstract

We analyze a coupled Cahn-Hilliard-Forchheimer system featuring concentration-dependent mobility, mass source and convective transport. The velocity field is governed by a generalized quasi-incompressible Forchheimer equation with solution-dependent volume source. We impose Dirichlet boundary conditions for the pressure to accommodate the source term. Our contributions include a novel well-posedness result for the generalized Forchheimer subsystem via the Browder-Minty theorem, and existence of weak solutions for the full coupled system established through energy estimates at the Galerkin level combined with compactness techniques such as Aubin-Lions' lemma and Minty's trick. Furthermore, we develop a structure-preserving discretization using Raviart-Thomas elements for the velocity that maintains exact mass balance and discrete energy-dissipation balance, with well-posedness demonstrated through relative energy estimates and inf-sup stability. Lastly, we validate our model through numerical experiments, demonstrating optimal convergence rates, structure preservation, and the role of the Forchheimer nonlinearity in governing phase-field evolution dynamics.