329 research outputs found
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
An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
The Cahn--Hilliard equation is a fundamental model that describes phase
separation processes of binary mixtures. In recent years, several types of
dynamic boundary conditions have been proposed in order to account for possible
short-range interactions of the material with the solid wall. Our first aim in
this paper is to propose a new class of dynamic boundary conditions for the
Cahn--Hilliard equation in a rather general setting. The derivation is based on
an energetic variational approach that combines the least action principle and
Onsager's principle of maximum energy dissipation. One feature of our model is
that it naturally fulfills three important physical constraints such as
conservation of mass, dissipation of energy and force balance relations. Next,
we provide a comprehensive analysis of the resulting system of partial
differential equations. Under suitable assumptions, we prove the existence and
uniqueness of global weak/strong solutions to the initial boundary value
problem with or without surface diffusion. Furthermore, we establish the
uniqueness of asymptotic limit as and characterize the stability
of local energy minimizers for the system.Comment: to appear in Arch. Rational Mech. Ana
Well-posedness and longtime behavior for the modified phase-field crystal equation
We consider a modification of the so-called phase-field crystal (PFC)
equation introduced by K.R. Elder et al. This variant has recently been
proposed by P. Stefanovic et al. to distinguish between elastic relaxation and
diffusion time scales. It consists of adding an inertial term (i.e. a
second-order time derivative) into the PFC equation. The mathematical analysis
of the resulting equation is more challenging with respect to the PFC equation,
even at the well-posedness level. Moreover, its solutions do not regularize in
finite time as in the case of PFC equation. Here we analyze the modified PFC
(MPFC) equation endowed with periodic boundary conditions. We first prove the
existence and uniqueness of a solution with initial data in a bounded energy
space. This solution satisfies some uniform dissipative estimates which allow
us to study the global longtime behavior of the corresponding dynamical system.
In particular, we establish the existence of an exponential attractor. Then we
demonstrate that any trajectory originating from the bounded energy phase space
does converge to a unique equilibrium. This is done by means of a suitable
version of the {\L}ojasiewicz-Simon inequality. A convergence rate estimate is
also given
Global well-posedness and attractors for the hyperbolic Cahn-Hilliard-Oono equation in the whole space
We prove the global well-posedness of the so-called hyperbolic relaxation of
the Cahn-Hilliard-Oono equation in the whole space R^3 with the non-linearity
of the sub-quintic growth rate. Moreover, the dissipativity and the existence
of a smooth global attractor in the naturally defined energy space is also
verified. The result is crucially based on the Strichartz estimates for the
linear Scroedinger equation in R^3
On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials
We consider a model describing the evolution of a tumor inside a host tissue
in terms of the parameters , (proliferating and dead
cells, respectively), (cell velocity) and (nutrient concentration). The
variables , satisfy a Cahn-Hilliard type system with
nonzero forcing term (implying that their spatial means are not conserved in
time), whereas obeys a form of the Darcy law and satisfies a
quasistatic diffusion equation. The main novelty of the present work stands in
the fact that we are able to consider a configuration potential of singular
type implying that the concentration vector is
constrained to remain in the range of physically admissible values. On the
other hand, in view of the presence of nonzero forcing terms, this choice gives
rise to a number of mathematical difficulties, especially related to the
control of the mean values of and . For the resulting
mathematical problem, by imposing suitable initial-boundary conditions, our
main result concerns the existence of weak solutions in a proper regularity
class.Comment: 41 page
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