138 research outputs found
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
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
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 -truncation method
we prove existence of weak solutions for a power-law exponent
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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
Short Time Existence for the Curve Diffusion Flow with a Contact Angle
We show short-time existence for curves driven by curve diffusion flow with a
prescribed contact angle : The evolving curve has free
boundary points, which are supported on a line and it satisfies a no-flux
condition. The initial data are suitable curves of class with
. For the proof the evolving curve is represented
by a height function over a reference curve: The local well-posedness of the
resulting quasilinear, parabolic, fourth-order PDE for the height function is
proven with the help of contraction mapping principle. Difficulties arise due
to the low regularity of the initial curve. To this end, we have to establish
suitable product estimates in time weighted anisotropic -Sobolev spaces of
low regularity for proving that the non-linearities are well-defined and
contractive for small times.Comment: 38 page
Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids
We introduce a new sharp interface model for the flow of two immiscible,
viscous, incompressible fluids. In contrast to classical models for two-phase
flows we prescribe an evolution law for the interfaces that takes diffusional
effects into account. This leads to a coupled system of Navier-Stokes and
Mullins-Sekerka type parts that coincides with the asymptotic limit of a
diffuse interface model. We prove the long-time existence of weak solutions,
which is an open problem for the classical two-phase model. We show that the
phase interfaces have in almost all points a generalized mean curvature.Comment: 26 page
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