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

TL;DR

This work analyzes a Cahn–Hilliard–Forchheimer system with solution-dependent mobility and mass/volume sources, motivated by tumor-growth modeling. It proves a new well-posedness result for the generalized Forchheimer subsystem via Browder–Minty and establishes existence of weak solutions for the coupled system using a Galerkin approach, energy estimates, and compactness arguments. A fully discrete, structure-preserving scheme based on Raviart–Thomas velocity spaces preserves exact mass balance and discrete energy dissipation and is shown to be well-posed through relative energy estimates and inf–sup stability. Numerical experiments confirm optimal convergence rates and structure preservation, and illustrate how the Forchheimer nonlinearity governs phase-field evolution and boundary interactions.

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.

Figures (8)

• Figure 1: Comparison of results for the generalized compressible Forchheimer system $\beta|{\bm{v}}|^{s-1}{\bm{v}}+\alpha {\bm{v}}+\nabla p = \bm{f}$ and $\textup{div} {\bm{v}} = g$.
• Figure 2: Phase-field $\phi$ evolution with $\gamma=0$ for $s \in \{1,2,3\}$ (top to bottom) at times $t\in\{0.4,0.8,1.2,1.6,2,5\}$ (left to right); mass leakage via Neumann-like conditions causes unphysical boundary layers with $\phi\geq1$.
• Figure 3: Phase-field $\phi$ evolution with $\gamma=1$ for $s \in \{1,2,3\}$ (top to bottom) at times $t\in\{0.4,0.8,1.2,1.6,2,5\}$ (left to right); boundary conditions enable mass influx, driving domain invasion.
• Figure 4: Evolution of the mass (left) and energy (right); strict mass increase ($\gamma=0$) vs. leakage ($\gamma=1$) and energy decay aligns with theoretical dissipation.
• Figure 5: Structure-preserving properties of the scheme; machine-precision mass balance errors (left) and non-positive energy dissipation error (right), confirming discrete entropy stability.

Theorems & Definitions (12)

• Remark 2.1
• Lemma 2.2
• Lemma 2.3: cf. galdi2011introduction
• Definition 3.2: Weak solution of \ref{['Eq:CHF']}
• Remark 3.3
• Theorem 3.4
• proof
• Lemma 4.1: Discrete interpolation inequalitiesDiegel2017Heywood1982
• Lemma 4.2: Inf-sup stabilitypan2012mixed
• Theorem 4.5