223 research outputs found
On a Cahn-Hilliard type phase field system related to tumor growth
The paper deals with a phase field system of Cahn-Hilliard type. For positive
viscosity coefficients, the authors prove an existence and uniqueness result
and study the long time behavior of the solution by assuming the nonlinearities
to be rather general. In a more restricted setting, the limit as the viscosity
coefficients tend to zero is investigated as well.Comment: Key words: phase field model, tumor growth, viscous Cahn-Hilliard
equations, well posedness, long-time behavior, asymptotic analysi
A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions
A boundary control problem for the viscous Cahn-Hilliard equations with
possibly singular potentials and dynamic boundary conditions is studied and
first order necessary conditions for optimality are proved.
Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase
separation, singular potentials, optimal control, optimality conditions,
adjoint state syste
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
Well-posedness and regularity for a generalized fractional Cahn-Hilliard system
In this paper, we investigate a rather general system of two operator
equations that has the structure of a viscous or nonviscous Cahn--Hilliard
system in which nonlinearities of double-well type occur. Standard cases like
regular or logarithmic potentials, as well as non-differentiable potentials
involving indicator functions, are admitted. The operators appearing in the
system equations are fractional versions of general linear operators and
, where the latter are densely defined, unbounded, self-adjoint and monotone
in a Hilbert space of functions defined in a smooth domain and have compact
resolvents. We remark that our definition of the fractional power of operators
uses the approach via spectral theory. Typical cases are given by standard
second-order elliptic operators (e.g., the Laplacian) with zero Dirichlet or
Neumann boundary conditions, but also other cases like fourth-order systems or
systems involving the Stokes operator are covered by the theory. We derive
general well-posedness and regularity results that extend corresponding results
which are known for either the non-fractional Laplacian with zero Neumann
boundary condition or the fractional Laplacian with zero Dirichlet condition.
It turns out that the first eigenvalue of plays an important
und not entirely obvious role: if is positive, then the operators
and may be completely unrelated; if, however, ,
then it must be simple and the corresponding one-dimensional eigenspace has to
consist of the constant functions and to be a subset of the domain of
definition of a certain fractional power of . We are able to show general
existence, uniqueness, and regularity results for both these cases, as well as
for both the viscous and the nonviscous system.Comment: 36 pages. Key words: fractional operators, Cahn-Hilliard systems,
well-posedness, regularity of solution
Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: a `deep quench' approach
In this paper, we investigate a distributed optimal control problem for a
convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such
systems govern phase separation processes between two phases taking place in an
incompressible fluid in a container and, at the same time, on the container
boundary. The cost functional is of standard tracking type, while the control
is exerted by the velocity of the fluid in the bulk. In this way, the coupling
between the state (given by the associated order parameter and chemical
potential) and control variables in the governing system of nonlinear partial
differential equations is bilinear, which presents a difficulty for the
analysis. In contrast to the previous paper arXiv:1709.02335 [math.AP] by the
same authors, the bulk and surface free energies are of double obstacle type,
which renders the state constraint nondifferentiable. It is well known that for
such cases standard constraint qualifications are not satisfied so that
standard methods do not apply to yield the existence of Lagrange multipliers.
In this paper, we overcome this difficulty by taking advantage of results
established in the quoted paper for logarithmic nonlinearities, using a
so-called `deep quench approximation'. We derive results concerning the
existence of optimal controls and the first-order necessary optimality
conditions in terms of a variational inequality and the associated adjoint
system.Comment: Key words: Cahn-Hilliard system, convection term, dynamic boundary
conditions, double obstacle potentials, optimal velocity control, optimality
conditions. arXiv admin note: text overlap with arXiv:1702.0190
On a Cahn-Hilliard system with convection and dynamic boundary conditions
This paper deals with an initial and boundary value problem for a system
coupling equation and boundary condition both of Cahn-Hilliard type; an
additional convective term with a forced velocity field, which could act as a
control on the system, is also present in the equation. Either regular or
singular potentials are admitted in the bulk and on the boundary. Both the
viscous and pure Cahn-Hilliard cases are investigated, and a number of results
is proven about existence of solutions, uniqueness, regularity, continuous
dependence, uniform boundedness of solutions, strict separation property. A
complete approximation of the problem, based on the regularization of maximal
monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and
rigorously discussed.Comment: Key words: Cahn-Hilliard system, convection, dynamic boundary
condition, initial-boundary value problem, well-posedness, regularity of
solution
Distributed optimal control of a nonstandard nonlocal phase field system
We investigate a distributed optimal control problem for a nonlocal phase
field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal
version of a model for two-species phase segregation on an atomic lattice under
the presence of diffusion that has been studied in a series of papers by P.
Podio-Guidugli and the present authors. The model consists of a highly
nonlinear parabolic equation coupled to an ordinary differential equation. The
latter equation contains both nonlocal and singular terms that render the
analysis difficult. 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.Comment: 38 Pages. Key words: distributed optimal control, nonlinear phase
field systems, nonlocal operators, first-order necessary optimality
condition
Singular limit of an integrodifferential system related to the entropy balance
A thermodynamic model describing phase transitions with thermal memory, in
terms of an entropy equation and a momentum balance for the microforces, is
adressed. Convergence results and error estimates are proved for the related
integrodifferential system of PDE as the sequence of memory kernels converges
to a multiple of a Dirac delta, in a suitable sense.Comment: Key words: entropy equation, thermal memory, phase field model,
nonlinear partial differential equations, asymptotics on the memory ter
Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential
This paper is concerned with a distributed optimal control problem for a
nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version
of a model for two-species phase segregation on an atomic lattice under the
presence of diffusion. The local model has been investigated in a series of
papers by P. Podio-Guidugli and the present authors; the nonlocal model studied
here consists of a highly nonlinear parabolic equation coupled to an ordinary
differential inclusion of subdifferential type. The inclusion originates from a
free energy containing the indicator function of the interval in which the
order parameter of the phase segregation attains its values. It also contains a
nonlocal term modeling long-range interactions. Due to the strong nonlinear
couplings between the state variables (which even involve products with time
derivatives), the analysis of the state system is difficult. In addition, the
presence of the differential inclusion is the reason that standard arguments of
optimal control theory cannot be applied to guarantee the existence of Lagrange
multipliers. In this paper, we employ recent results proved for smooth
logarithmic potentials and perform a so-called `deep quench' approximation to
establish existence and first-order necessary optimality conditions for the
nonsmooth case of the double obstacle potential.Comment: Key words: distributed optimal control, phase field systems, double
obstacle potentials, nonlocal operators, first-order necessary optimality
conditions. The interested reader can also see the related preprints
arXiv:1511.04361 and arXiv:1605.07801 whose results are recalled and used for
the analysis carried out in this pape
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