169 research outputs found
On the longtime behavior of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions
In this paper, we study the longtime asymptotic behavior of a phase
separation process occurring in a three-dimensional domain containing a fluid
flow of given velocity. This process is modeled by a viscous convective
Cahn-Hilliard system, which consists of two nonlinearly coupled second-order
partial differential equations for the unknown quantities, the chemical
potential and an order parameter representing the scaled density of one of the
phases. In contrast to other contributions, in which zero Neumann boundary
conditions were are assumed for both the chemical potential and the order
parameter, we consider the case of dynamic boundary conditions, which model the
situation when another phase transition takes place on the boundary. The phase
transition processes in the bulk and on the boundary are driven by free
energies functionals that may be nondifferentiable and have derivatives only in
the sense of (possibly set-valued) subdifferentials. For the resulting
initial-boundary value system of Cahn-Hilliard type, general well-posedness
results have been established in a recent contribution by the same authors. In
the present paper, we investigate the asymptotic behavior of the solutions as
times approaches infinity. More precisely, we study the -limit (in a
suitable topology) of every solution trajectory. Under the assumptions that the
viscosity coefficients are strictly positive and that at least one of the
underlying free energies is differentiable, we prove that the -limit is
meaningful and that all of its elements are solutions to the corresponding
stationary system, where the component representing the chemical potential is a
constant.Comment: Key words: Cahn-Hilliard systems, convection, dynamic boundary
conditions, well-posedness, asymptotic behavior, omega-limit. arXiv admin
note: text overlap with arXiv:1704.0533
Optimal distributed control of a generalized fractional Cahn-Hilliard system
In the recent paper `Well-posedness and regularity for a generalized
fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors,
general well-posedness results have been established for a a class of
evolutionary systems of two equations having the structure of a viscous
Cahn-Hilliard system, in which nonlinearities of double-well type occur. The
operators appearing in the system equations are fractional versions in the
spectral sense of general linear operators A,B having compact resolvents, which
are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of
functions defined in a smooth domain. In this work we complement the results
given in arXiv:1804.11290 by studying a distributed control problem for this
evolutionary system. The main difficulty in the analysis is to establish a
rigorous Frechet differentiability result for the associated control-to-state
mapping. This seems only to be possible if the state stays bounded, which, in
turn, makes it necessary to postulate an additional global boundedness
assumption. One typical situation, in which this assumption is satisfied,
arises when B is the negative Laplacian with zero Dirichlet boundary conditions
and the nonlinearity is smooth with polynomial growth of at most order four.
Also a case with logarithmic nonlinearity can be handled. Under the global
boundedness assumption, we establish existence and first-order necessary
optimality conditions for the optimal control problem in terms of a variational
inequality and the associated adjoint state system.Comment: Key words: fractional operators, Cahn-Hilliard systems, optimal
control, necessary optimality condition
Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
The paper arXiv:1804.11290 contains well-posedness and regularity results for
a system of evolutionary operator equations having the structure of a
Cahn-Hilliard system. The operators appearing in the system equations were
fractional versions in the spectral sense of general linear operators A and B
having compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process modeled
by the Cahn-Hilliard system could be of a very general type that includes
standard physically meaningful cases such as polynomial, logarithmic, and
double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an
analysis of distributed optimal control problems was performed for such
evolutionary systems, where only the differentiable case of certain polynomial
and logarithmic double-well potentials could be admitted. Results concerning
existence of optimizers and first-order necessary optimality conditions were
derived. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called "deep quench" method. We first
give a general convergence analysis of the deep quench approximation that
includes an error estimate and then demonstrate that its use leads in the
double obstacle case to appropriate first-order necessary optimality conditions
in terms of a variational inequality and the associated adjoint state system.Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal
control, double obstacles, necessary optimality condition
Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
In this paper, we study an optimal control problem for a two-dimensional
Cahn-Hilliard-Darcy system with mass sources that arises in the modeling of
tumor growth. The aim is to monitor the tumor fraction in a finite time
interval in such a way that both the tumor fraction, measured in terms of a
tracking type cost functional, is kept under control and minimal harm is
inflicted to the patient by administering the control, which could either be a
drug or nutrition. We first prove that the optimal control problem admits a
solution. Then we show that the control-to-state operator is Fr\'echet
differentiable between suitable Banach spaces and derive the first-order
necessary optimality conditions in terms of the adjoint variables and the usual
variational inequality
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