Abstracts of Publications:
Abstract:
We consider the physical
stability of an imbedded minimal surface $\Phi$. Assuming that
Abstract:
We consider the free
boundary problem of a liquid drop with viscosity and surface tension.
We study the
linearized equations with semigroup
methods to get existence results for the nonlinear problem. The
spectrum of the
generator is computed. Large surface tension creates nonreal
eigenvalues and an exterior force results
in a Hopf bifurcation. The methods are
used to study
wind-generated surface waves.
Abstract:
We analyze the evolution problem of a body of liquid with a free boundary and surface tension in two space dimensions. We prove with direct methods a priori estimates and the existence of a weak solution on a short time interval. We use Hilbert spaces that correspond to energies and that are appropriate for numerical purpose. A stable Galerkin scheme is constructed.
Abstract:
We study the free boundary
between a viscous fluid and an inviscid
fluid
satisfying the Navier-Stokes and Euler
equations
respectively. Surface tension is incorporated. We read the equations as
a free
boundary problem for one viscous fluid with a nonlocal
boundary force. A decomposition of the pressure distribution in the inviscid fluid identifies the main part of the
generator.
It is used to derive a short time existence result for the two-phase
problem.
Abstract:
We consider fluid systems
with a free boundary and with a point of contact of the free boundary
with a
solid wall. We contribute to the discussion on the conditions for the
dynamic
contact angle and well-posedness of the
equations. An energy equality suggests a
constant angle. With the help
of symmetric extensions we prove a resolvent
estimate
in the case of a 90 degree contact angle. A technique developed by Renardy can be applied and yields an existence
result for
the nonlinear problem.
Abstract:
The stationary Stokes
equations with a free boundary are studied in a perforated domain. The
perforation consists of a periodic array of cylinders of size and
distance $O(\eps)$.
The free boundary is given
as the graph of a function on a two-dimensional perforated domain. We
derive
equations for the two-scale limit of solutions. The limiting equation
is a free
boundary system. It involves a nonlinear elliptic operator
corresponding to the
nonlinear mean-curvature expression in the original equations.
Depending on the
equation for the contact-angle the pressure is in general unbounded.
Abstract:
We analyze two partial
differential equations that are posed on perforated domains. We provide
estimates for the solutions, that do not
depend on the
size of the perforation. The first problem concerns homogenization of
the
Laplace- and the mean-curvature operator with Neumann boundary
conditions. We
derive uniform Lipschitz-estimates for the
solutions.
The result is used in the analysis of a free boundary system of fluid
mechanics. A contractive iteration yields the existence of solutions
and
uniform estimates. The key is the use of function spaces that are
different
from the usual $L^p$-spaces.
Abstract:
We study the generation of
surface waves on water as a bifurcation phenomenon. For a critical wind
speed
there appear traveling wave solutions.
While the
linear waves do not correspond to a mass-transport, nonlinear effects
create a
shear flow and result in a net mass transport in the direction of the
wind. We
derive an asymptotic formula for the average tangential velocity along
the free
surface. Numerical investigations confirm the appearance of the shear
flow and
yield results on the direction of the bifurcation.
Abstract:
We present a time discretization
for the single phase Stefan problem with
Gibbs-Thomson law. The method resembles an operator splitting scheme
with an
evolution step for the temperature distribution and a transport step
for the
dynamics of the free boundary. The evolution step only involves the
solution of
a linear equation that is posed on the old domain. We prove that the
proposed
scheme is stable in function spaces of high regularity. In the limit
$\Delta
t\to 0$ we find strong solutions of the continuous problem. This proves
consistency of the scheme and it additionally yields a new short-time
existence
result for the continuous problem.
Abstract:
In models for two-phase
flow in porous media one imposes that the microscopic pressures in the
two
fluids differ by a constant, the capillary pressure. We investigate the
behavior of a microscopic interphase
between the two fluids in order to derive an expression for the
capillary
pressure. We find that the averaged equations are instationary.
In limit cases a constant capillary pressure can be deduced; its value
depends
on the flow direction and reproduces a well-known hysteresis
effect. Three models are studied, a deterministic, a stochastic, and a
stochastic Hele-Shaw free boundary model.
The upscaled equations are derived in the
deterministic case
under a condition on typical evolution patterns. In the stochastic
cases the
limit equations hold almost surely.
Abstract:
We study the dependence of
the Hausdorff measure $\H1_d$ on the
distance $d$. We
show that the uniform convergence of $d_j$
to $d$ is
equivalent to the $\Gamma$ convergence of $\H1_{d_j}$ to $\H1_d$ with respect to the Hausdorff
convergence on compact connected subsets. We also consider the case
when
distances are replaced by semi-distances.
Abstract:
We study the equations for
an incompressible ideal fluid with a free surface that is subject to
surface
tension; it is not assumed that the fluid is irrotational.
We derive a priori estimates for smooth solutions and prove a
short-time
existence result. The estimates are based on a careful study of the
regularity
properties of the pressure function. An adequate artificial coordinate
system
is used to replace the standard Lagrangian
coordinates. The solutions to the Euler equations are obtained as
vanishing
viscosity limits of solutions of an appropriate Navier-Stokes
system.
Abstract:
We obtain a Gamma-convergence result for the gradient theory of solid-solid phase transitions, in the case of two geometrically linear wells in two dimensions. We consider the functionals \[ I_\e[u] = \int_\Omega \frac{1}{\e} W(\nabla u) + \e |\nabla^2u|^2\] where $u:\Omega\subset\R^2\to\R^2$, $W$ depends only on the symmetric part of $\nabla u$, and $W(F)=0$ for two distinct values of $F$, say $A$ and $B$. We show that, under suitable growth assumptions on $W$ and for star-shaped domains $\Omega$, as $\e\to0$ $I_\e$ converges, in the sense of Gamma convergence, to a functional $I_0$. The limit $I_0$ is finite only on functions $u$ such that the symmetric part of $\nabla u$ is a function of bounded variation which takes only values $A$ and $B$. On those functions, the energy concentrates on the jump set $J$ of $\nabla u$, with a surface energy depending on the normal $\nu$ to $J$, and is given by \[I_0[u]= \int_{J} k(\nu) d\calH^1\,. \] The interfaces can have, in general, two orientations.
Abstract:
We study systems of
reaction diffusion type for two species in one space dimension and
investigate
the dynamics in the case that the second species does not diffuse. We
consider
competing species with two stable equilibria
and
front solutions that connect the two stable states. A free energy
function
determines a preferred state. If the diffusive species is preferred, traveling waves may appear. Instead, if the
non-diffusive
species is preferred, stationary fronts are the only monotone traveling waves. We show that these fronts are
unstable and
that the non-diffusive species can propagate at a logarithmic rate.
Abstract:
The singularly perturbed
two-well problem in the theory of solid-solid phase transitions takes
the form
\[ I_\e[u] = \int_\Omega \frac{1}{\e}
W(\nabla u) + \e |\nabla^2u|^2,\] where
$u:\Omega\subset\R^n\to\R^n$ is the deformation, and $W$ vanishes for
all
matrices in $K=SO(n)A \cup SO(n)B$. We focus on the case $n=2$ and
derive, by
means of Gamma convergence, a sharp-interface limit for $I_\e$. The
proof is
based on a rigidity estimate for low-energy functions. Our rigidity
argument
also gives an optimal two-well Liouville
estimate: if
$\nabla u$ has a small $BV$ norm (compared
to the
diameter of the domain), then, in the $L^1$ sense, either the distance
of $\nabla u$ from $SO(2)A$
or the one
from $SO(2)B$ is controlled by the distance of $\nabla
u$ from $K$. This implies that the oscillation of $\nabla
u$ in weak-$L^1$ is controlled by the $L^1$ norm of the distance of $\nabla u$ to $K$.
Abstract:
Fluids in unsaturated
porous media are described by the pressure $p$ and the saturation
$u$.
Darcy's law and conservation of mass provides an evolution equation for
$u$,
the capillary pressure provides a relation between $p$ and $u$ of the
form
$p\in p_c(u,\del_t u)$. The multi-valued
function $p_c$ leads to hysteresis
effects. We construct weak and strong solutions to the hysteresis system and homogenize the system for
oscillatory
stochastic coefficients. The effective equations contain a new
dependent
variable which encodes the history of the wetting process and provide a
better
description of the physical system.
Abstract:
We study the family of functionals \[ I_\e[u] = \int_\Omega \frac{1}{\e} W(\nabla u) + \e |\nabla^2 u|^2\,dx,\] with $u:\Omega\subset\R^n\to\R^n$ representing the deformation of an elastic body, and $W$ the energy density, which vanishes for all matrices in $K=SO(n)A \cup SO(n)B$. The energy $I_\e$ describes an elastic material with two preferred gradients and surface tension, the so-called two-well problem of solid-solid phase transitions. The Gamma limit of the functionals $I_\e$ was determined, for $n=2$, in [ContiSchweizer1], the crucial step in the proof is to derive rigidity estimates in order to control the local rotations of minimizing sequences. While [ContiSchweizer1] treats the case that $W$ has quadratic growth at infinity, we treat here the case that $W$ does not permit self-penetration, i.e. $W(F) = \infty$ for $\det F < 0$. We restrict to $n=2$ and exploit results of [ContiSchweizer1].
Abstract:
We
consider a nonlinear one-dimensional scalar equation of diffusion type
in
which, depending on the gradient of the solution, the diffusion
coefficient may
be positive or negative. We compare two concepts of Young measure
solutions
which are based on different methods to construct approximate
solutions, the
SP-solutions (singular perturbation) and the EM-solution (energy
minimization).
We show that the SP-solution can recover classical solutions where the
EM-solution fails to do so, and that EM-solutions are more stable under
perturbations of the initial values. We characterize the EM-solution
with a
free boundary problem and determine its long-time behavior.
Abstract:
We study a porous medium
with saturated, unsaturated, and dry regions, described by Richards'
equation
for the saturation $s$ and the pressure $p$. Due to a degenerate
permeability
coefficient $k(x,s)$
and a
degenerate capillary pressure function $p_c(x,s)$, the equations may be of elliptic,
parabolic, or of
ODE-type. We construct a parabolic regularization of the equations and
find
conditions that guarantee the convergence of the parabolic solutions to
a
solution of the degenerate system. An example shows that the
convergence fails
in general. Our approach provides an existence result for the outflow
problem
in the case of $x$-dependent coefficients, and a method for a numerical
approximation.
Abstract:
We consider the
one-dimensional degenerate two-phase flow equations as a model for
water-drive
in oil recovery. The effect of oil trapping is observed in strongly
heterogeneous materials with large variations in the permeabilities
and in the capillary pressure curves. In such materials, a
vanishing oil saturation may appear at interior interfaces and
inhibit
the oil recovery. We introduce a free boundary problem that separates a
critical region with vanishing permeabilities
from a
strictly parabolic region and give a rigorous derivation of the
effective
conservation law.
Abstract:
We derive interior $L^p$-estimates
for solutions of linear elliptic systems
with oscillatory coefficients. The estimates are independent of $\eps$, the small length scale of the rapid
oscillations. So
far, such results are based on potential theory and restricted to
periodic
coefficients. Our approach relies on BMO-estimates and an interpolation
argument, gradients are treated with the
help of finite
differences. This allows to treat
coefficients that
depend on a fast and a slow variable. The estimates imply an $L^p$-corrector result for approximate solutions.
Abstract:
We study a pore scale model
for the catalyst layer on the cathod e
side of a fuel
cell, where hydrogen and oxygen combine at catalyst sites. Our model
distinguishes microscopically the phases of rigid structure,
electrolyte,
pore-space, and catalyst. The oxygen concentration and the protonic
potential are described by diffu sion equations with
reaction terms on the catalyst's
surface. For the limit of a vanishing pore size we derive homogenized
equations
of reaction-diffusion type and provide formulae for the effective coeffici ents. A
dimensional
reduction shows that a thin catalyst layer can be replaced by a
boundary
condition. We furthermore analyze the effect of a doubling of the Tafel slope for high protonic
potentials and determine effective constants.
Abstract:
We propose a new method for
the homogenization of hysteresis models of
plasticity. For the one-dimensional wave equation with an elasto-plastic
stress-strain relation we derive averaged equations and perform the
homogenization limit for stochastic material parameters. This
generalizes
results of the seminal paper by Franc{\ocirc{u}} and Krej{\v{c}}{\'{\i}}. Our approach rests on energy methods
for partial
differential equations and provides short proofs without recurrence to hysteresis operator theory. It has the potential
to be extended
to the higher dimensional case.
Abstract:
We introduce an
approximation procedure and provide existence results for two-phase
flow
equations in porous media. The medium can have hydrophobic and
hydrophilic
components such that the capillary pressure function is degenerate for
extreme
saturations. Our main interest is the outflow boundary condition which
models
an interface with open space. The approximate system introduces
standard
boundary conditions and can be used in numerical schemes. It allows the
derivation of maximum principles. These are the basis for the
derivation of the
limiting system in the form of a variational
inequality.
Abstract:
We analyze the time
harmonic Maxwell's equations in a complex geometry. The scatterer
$\Omega\subset\R^3$ contains a periodic pattern of small wire
structures of
high conductivity, the single element has the shape of a split ring. We
rigorously derive effective equations for the scatterer
and provide formulas for the effective permittivity and permeability.
The
latter turns out to be frequency dependent and has
a
negative real part for appropriate parameter values. This
magnetic
activity is the key feature of a left-handed meta-material.
Abstract:
We study the two-phase flow
equations describing, e.g., the motion of oil and water in a porous
material,
and are concerned with interior interfaces where two different porous
media are
in contact. At such an interface, the entry pressure relation together
with the
degeneracy of the system leads to an interesting effect known as
oil-trapping.
Restricting to the one-dimensional case we show an existence result
with the
help of appropriate regularizations and a time discretization. The crucial tool is
a
compactness lemma: The control of the time derivative in a space of
measures is
used to conclude the strong convergence of a sequence.
Mathematical
Methods in the Applied Sciences, 33, 974-984, 2010
Abstract:
The
catalyst layer in a fuel cell can be described with a system of
reaction
diffusion equations for the oxygen concentration and the protonic
overpotential. The Tafel
law
gives an exponential expression for the reaction rate, and the Tafel slope is a coefficient in this law. We
present a
rigorous thin layer analysis for two reaction regimes. In the case of
thin
catalyst layers and bounded potentials, the original Tafel
law enters as an effective boundary condition. Instead, in the case of
large protonic overpotentials,
we
derive an exponential law which contains the doubled Tafe
slope.
Abstract:
A Riemannian metric $a$ in the plane together with a point $A\subset \R^2$ induces a corresponding distance function $d_a(A,.)$. We investigate optimization problems that seek for a metric $a$ such that the distance function is large, either the distance between $A$ and a set $B$ or an integrated distance such that we maximize the Wasserstein distance between $A$ and the set $B$. We find necessary conditions for optimal metrics which help to determine solutions. In the case that the set $B$ is a single point, we determine the optimal metric explicitely.
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