Strong Well-Posedness of a Diffuse Interface Model for a Viscous, Quasi-Incompressible Two-Phase Flow

We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study a model for the general case that the fluids have different densities due to Lowengrub and Truskinovski. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. We prove existence of unique strong solutions for the non-stationary system for sufficiently small times.Comment: 30 page

On a Model for Phase Separation on Biological Membranes and its Relation to the Ohta-Kawasaki Equation

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, R\"atz, R\"oger and the second author. The model is an extended Cahn-Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta-Kawasaki equation as the limit for infinitely large affinity between membrane components.Comment: 41 page

Spectral Invariance of Non-Smooth Pseudodifferential Operators

In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in H\"older spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in H\"older spaces, we improve the characterization of non-smooth pseudodifferential operators in a previous work by the authors.Comment: 43 page

Weak Solutions for a Non-Newtonian Diffuse Interface Model with Different Densities

We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier-Stokes system and a Cahn-Hilliard equation. For the Cahn-Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the $L^\infty$-truncation method we prove existence of weak solutions for a power-law exponent $p>\frac{2d+2}{d+2}$, $d=2,3$

On Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows

We discuss the sharp interface limit of a diffuse interface model for a two-phase flow of two partly miscible viscous Newtonian fluids of different densities, when a certain parameter \epsilon>0 related to the interface thickness tends to zero. In the case that the mobility stays positive or tends to zero slower than linearly in \epsilon we will prove that weak solutions tend to varifold solutions of a corresponding sharp interface model. But, if the mobility tends to zero faster than \epsilon^3 we will show that certain radially symmetric solutions tend to functions, which will not satisfy the Young-Laplace law at the interface in the limit.Comment: 27 pages, 1 figur

Well-Posedness of a Navier-Stokes/Mean Curvature Flow system

We consider a two-phase flow of two incompressible, viscous and immiscible fluids which are separated by a sharp interface in the case of a simple phase transition. In this model the interface is no longer material and its evolution is governed by a convective mean curvature flow equation, which is coupled to a two-phase Navier-Stokes equation with Young-Laplace law. The problem arises as a sharp interface limit of a diffuse interface model, which consists of a Navier-Stokes system coupled with an Allen-Cahn equation. We prove existence of strong solutions for sufficiently small times and regular initial data.Comment: 32 page