35K55 Nonlinear parabolic equations
This paper deals with the analysis of an instationary drift-diffusion model for organic semicon- ductor devices including Gauss–Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent ener- getic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength.
To prove the existence of global weak solutions, we consider a problem with (for small den- sities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level en- sure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densi- ties. The estimates rely on Moser iteration techniques.
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.
In these notes we discuss two approaches to evolutionary Γ- convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ- convergence methods. In the first approach we consider families of uni- formly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savar ́e 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.
We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
Complete damage in linear elastic materials — Modeling, weak formulation and existence results
(2013)
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying
Dirichlet boundary conditions. The evolution of the system is
described by a doubly nonlinear differential inclusion for the damage
process and a degenerating quasi-static balance equation for the displacement field
which are strongly nonlinearly coupled.
In our proposed model, the material
may completely disintegrate which is indispensable for a realistic modeling of
damage processes in elastic materials. Complete damage theories
lead to several mathematical problems since, for instance, coercivity properties
of the free energy are lost and, therefore, several difficulties arise.
For the introduced complete damage model, we propose a classical
formulation and a corresponding suitable weak formulation in an
$SBV$-framework. The main aim is to prove existence of weak solutions
for the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete damage
models of this type.
For the existence results, we had to handle the following problem:
During the damage process it might occur that not completely damaged material regions are isolated
from the Dirichlet boundary. In this case, the
deformation field cannot be controlled in the transition from incomplete
to complete damage. To tackle this problem, we consider the evolution
process on a time-depending domain. In this context, two major challenges
arise:
Firstly, the time-dependent domain approach leads to jumps in the energy
which have to be accounted for in the energy inequality of the notion of
weak solutions. To handle this problem, several energy estimates are established
by $\Gamma$-convergence techniques. Secondly, the time-depending domain
might have bad smoothness properties such that Korn's inequality cannot be
applied. To this end, a covering result for such sets with smooth
compactly embedded domains has been shown.
We investigate a distributed optimal control problem for a phase field
model of Cahn-Hilliard type. The model describes two-species phase segregation
on an atomic lattice under the presence of diffusion; it has been introduced recently in
[4], on the basis of the theory developed in [15], and consists of a system of two
highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply
directly, although the control constraints and the cost functional are of standard type.
We show that the problem admits a solution, and we derive the first-order
necessary conditions of optimality.
We investigate a nonstandard phase field
model of Cahn-Hilliard type. The model, which was introduced in
[16], describes two-species phase segregation and consists of a
system of two highly nonlinearly coupled PDEs. It has been studied
recently in
[5], [6] for the case of homogeneous Neumann
boundary conditions. In this paper, we investigate the case that the
boundary condition for one of the unknowns of the system is of third
kind and nonhomogeneous. For the resulting system, we show
well-posedness, and we study optimal boundary control
problems. Existence of optimal controls is shown, and the first-order
necessary optimality conditions are derived. Owing to the strong
nonlinear couplings in the PDE system, standard arguments of optimal
control theory do not apply directly, although the control constraints
and the cost functional will be of standard type.
This paper is concerned with a diffusion model of phase-field type, consisting
of a {parabolic} system of two partial differential equations{,} interpreted as balances
of microforces and microenergy{, for two unknowns: the problem's order parameter $\rho$}
and the chemical potential $\mu$; each equation includes a viscosity term -- respectively, $\varepsilon \,\partial_t\mu$ and $\delta\,\partial_t\rho$ -- with $\varepsilon$ and $\delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper \cite{CGPS3}, we proved that this problem is \wepo\ and investigated the \loti\ \bhv\ of its $(\varepsilon,\delta)-$solutions. Here we discuss the asymptotic limit of the system as $\eps$
tends to $0$. We prove convergence of
$(\varepsilon,\delta)-$solutions to the corresponding solutions for
the case $\eps =0$, whose long-time behavior we characterize; in the
proofs, we employ compactness and monotonicity arguments.
We introduce nonsmooth Schur-Newton methods for the solution of the nonlinear discrete saddle-point problems arising from discretized vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The discrete problems are obtained by semi-implicit discretization in time and a first order finite element discretization in space. We incorporate the linear constraints that enforce solutions to stay on the Gibbs simplex using Lagrangian multipliers and prove existence of these multipliers under the assumption of a non-trivial initial condition for the order parameters.
In this paper we formulate a boundary layer approximation
for an Allen--Cahn-type equation involving a small parameter $\eps$. Here, $\eps$ is related to the thickness of the boundary layer and we are interested in the limit when $\eps$ tends to $0$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen--Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
Existence result for a class of generalized standard materials with thermomechanical coupling
(2012)
This paper deals with the study of a three-dimensional model of thermomechanical coupling for viscous solids exhibiting hysteresis effects. This model is written in accordance with the formalism of generalized standard materials. It is composed by the momentum equilibrium equation combined with the flow rule, which describes some stress-strain dependance, and the heat-transfer equation. An existence result for this thermodynamically consistent problem is obtained by using a fixed-point argument and some qualitative properties of the solutions are established.
A typical phase field approach for describing phase separation and
coarsening phenomena in alloys is the Cahn-Hilliard model. This
model has been generalized to the so-called Cahn-Larche system
by combining it with elasticity to capture non-neglecting deformation
phenomena,
which occur during phase separation and coarsening processes in the material.
In order to account for damage effects,
we extend the existing framework of Cahn-Hilliard and Cahn-Larche
systems by incorporating an internal damage variable of local character. This
damage variable allows to model the effect that damage of a material point is influenced by its local surrounding.
The damage process is described by a unidirectional rate-dependent evolution inclusion
for the internal variable.
For the introduced Cahn-Larche systems
coupled with rate-dependent
damage processes, we
establish a suitable notion of weak solutions and prove existence of weak solutions.
This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.
We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
This paper is concerned with the state-constrained optimal control of the
two-dimensional thermistor problem, a quasi-linear coupled system
of a parabolic and elliptic PDE with mixed boundary conditions.
This system models the heating of a conducting material by means of direct current.
Existence, uniqueness and continuity for the state system are derived by employing
maximal elliptic and parabolic regularity. By similar arguments the
linearized state system is discussed, while the adjoint system involving measures
is investigated using a duality argument. These results allow to derive
first-order necessary conditions for the optimal control problem.
A general abstract approximation scheme for rate-independent processes in the
energetic formulation is proposed and its convergence is proved under various
rather mild data qualifications. The abstract theory is illustrated on several
examples: plasticity with isotropic hardening, damage, debonding,
magnetostriction, and two models of martensitic transformation in
shape-memory alloys.
In this paper, we study an optimal control problem for a singular system of partial differential equations that models a nonisothermal phase transition with a nonconserved order parameter. The control acts through a third boundary condition for the absolute temperature and plays the role of the outside temperature. It is shown that the corresponding control-to-state mapping is well defined, and the existence of an optimal control and the first-order optimality conditions for a quadratic cost functional of Bolza type are established.