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

¹ Institute of Mathematics, Johannes Gutenberg University, Mainz, Germany
² Computational Methods for PDEs, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria

^* Corresponding author: marvin.fritz@ricam.oeaw.ac.at

Received: 14 April 2025
Accepted: 19 September 2025

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.