Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes
M. Hassan Farshbaf-Shaker Harald Garcke
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
Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis
Harald Garcke Kei Fong Lam

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
On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities
John W. Barrett Harald Garcke Robert Nürnberg
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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