DCDS-S
Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes
M. Hassan Farshbaf-Shaker Harald Garcke
Discrete & Continuous Dynamical Systems - S 2011, 4(2): 371-389 doi: 10.3934/dcdss.2011.4.371
In this paper we derive thermodynamically consistent higher order phase field models for the dynamics of biomembranes in incompressible viscous fluids. We start with basic conservation laws and an appropriate version of the second law of thermodynamics and obtain generalizations of models introduced by Du, Li and Liu [3] and Jamet and Misbah [11]. In particular we derive a stress tensor involving higher order derivatives of the phase field and generalize the classical Korteweg capillarity tensor.
keywords: fluid interfaces convection second law of thermodynamics dissipation inequality Phase field model weak solution. biomembrane Navier-Stokes equation momentum equation bending elastic energy
DCDS
Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis
Harald Garcke Kei Fong Lam
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4277-4308 doi: 10.3934/dcds.2017183

We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New difficulties are encountered due to the higher polynomial growth, but for regular potentials, we retain the continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms as established in the previous work. Furthermore, we deduce the well-posedness of a variant of the model with quasi-static nutrient by rigorously passing to the limit where the ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is small.

keywords: Tumor growth phase field model Cahn–Hilliard equation reactiondiffusion equations chemotaxis weak solutions Dirichlet boundary conditions well-posedness asymptotic analysis singular potentials
CPAA
Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants
Helmut Abels Harald Garcke Josef Weber
Communications on Pure & Applied Analysis 2019, 18(1): 195-225 doi: 10.3934/cpaa.2019011

Two-phase flow of two Newtonian incompressible viscous fluids with a soluble surfactant and different densities of the fluids can be modeled within the diffuse interface approach. We consider a Navier-Stokes/Cahn-Hilliard type system coupled to non-linear diffusion equations that describe the diffusion of the surfactant in the bulk phases as well as along the diffuse interface. Moreover, the surfactant concentration influences the free energy and therefore the surface tension of the diffuse interface. For this system existence of weak solutions globally in time for general initial data is proved. To this end a two-step approximation is used that consists of a regularization of the time continuous system in the first and a time-discretization in the second step.

keywords: Two-phase flow diffuse interface model variable surface tension surfactants global existence implicit time discretization Navier-Stokes equations Cahn-Hilliard equation
DCDS-S
On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities
John W. Barrett Harald Garcke Robert Nürnberg
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 1-14 doi: 10.3934/dcdss.2008.1.1
Using formally matched asymptotic expansions we identify the sharp interface asymptotic limit of an Allen-Cahn/Cahn-Hilliard system using a novel approach which enables us to handle the case of variational inequalities.
keywords: matched asymptotic expansions Degenerate Cahn--Hilliard/Allen--Cahn variational inequality degenerate parabolic problem.

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