81 research outputs found
Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system
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.Comment: Key words: Cahn-Hilliard equation, phase field model, well-posedness,
long-time behavio
On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation
This paper investigates a nonlocal version of a model for phase separation on
an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55
(2006) 105-118. The model consists of an initial-boundary value problem for a
nonlinearly coupled system of two partial differential equations governing the
evolution of an order parameter and the chemical potential. Singular
contributions to the local free energy in the form of logarithmic or
double-obstacle potentials are admitted. In contrast to the local model, which
was studied by P. Podio-Guidugli and the present authors in a series of recent
publications, in the nonlocal case the equation governing the evolution of the
order parameter contains in place of the Laplacian a nonlocal expression that
originates from nonlocal contributions to the free energy and accounts for
possible long-range interactions between the atoms. It is shown that just as in
the local case the model equations are well posed, where the technique of
proving existence is entirely different: it is based on an application of
Tikhonov's fixed point theorem in a rather unusual separable and reflexive
Banach space.Comment: The paper is dedicated to our friend Paolo Podio-Guidugli on the
occasion of his 75th birthday with best wishe
Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions
This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases
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
Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system
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.Comment: Key words: Cahn-Hilliard equation, phase field model, time
discretization, convergence, error estimate
Longtime behavior for a generalized Cahn-Hilliard system with fractional operators
In this contribution, we deal with the longtime behavior of the solutions to
the fractional variant of the Cahn-Hilliard system, with possibly singular
potentials, that we have recently investigated in the paper `Well-posedness and
regularity for a generalized fractional Cahn-Hilliard system' (see
arXiv:1804.11290). More precisely, we study the omega-limit of the phase
parameter and characterize it completely. Our characterization depends on the
first eigenvalue of one of the operators involved: if it is positive, then the
chemical potential vanishes at infinity and every element of the omega-limit is
a stationary solution to the phase equation; if, instead, the first eigenvalue
is 0, then every element of the omega-limit satisfies a problem containing a
real function related to the chemical potential. Such a function is nonunique
and time dependent, in general, as we show by an example. However, we give
sufficient conditions in order that this function be uniquely determined and
constant.Comment: Key words: Fractional operators, Cahn-Hilliard systems, longtime
behavio
Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition
In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies
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
Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition
In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies
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