404 research outputs found
Some remarks on stability for a phase-field model with memory
The phase field system with memory can be viewed as a phenomenological extension of the classical phase equations in which memory effects have been taken into account in both fields. Such memory effects could be important for example during phase transition in polymer melts in the proximity of the glass transition temperature where configurational degrees of freedom in the polymer melt constitute slowly relaxing "internal modes" which are di±cult to model explicitly. They should be relevant in particular to glass-liquid-glass transitions where re-entrance effects have been recently reported [27]. We note that in numerical studies based on sharp interface equations obtained from (PFM), grains have been seen to rotate as they shrink [35, 36]. While further modelling and numerical efforts are now being undertaken, the present manuscript is devoted to strengthening the analytical underpinnings of the model
Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections
We consider a stochastic partial differential equation with two logarithmic
nonlinearities, with two reflections at 1 and -1 and with a constraint of
conservation of the space average. The equation, driven by the derivative in
space of a space-time white noise, contains a bi-Laplacian in the drift. The
lack of the maximum principle for the bi-Laplacian generates difficulties for
the classical penalization method, which uses a crucial monotonicity property.
Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we obtain
existence and uniqueness of solution for initial conditions in the interval
. Finally, we prove that the unique invariant measure is ergodic, and
we give a result of exponential mixing
Existence and stability of singular patterns in a GinzburgâLandau equation coupled with a mean field
We study singular patterns in a particular system of parabolic partial differential equations which consist of a GinzburgâLandau equation and a mean field equation. We prove the existence of the three simplest concentrated periodic stationary patterns (single spikes, double spikes, double transition layers) by composing them of more elementary patterns and solving the corresponding consistency conditions. In the case of spike patterns we prove stability for sufficiently large spatial periods by first showing that the eigenvalues do not tend to zero as the period goes to infinity and then passing in the limit to a nonlocal eigenvalue problem which can be studied explicitly. For the two other patterns we show instability by using the variational characterization of eigenvalues
Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term
The paper is devoted to a modification of the classical Cahn-Hilliard
equation proposed by some physicists. This modification is obtained by adding
the second time derivative of the order parameter multiplied by an inertial
coefficient which is usually small in comparison to the other physical
constants. The main feature of this equation is the fact that even a globally
bounded nonlinearity is "supercritical" in the case of two and three space
dimensions. Thus the standard methods used for studying semilinear hyperbolic
equations are not very effective in the present case. Nevertheless, we have
recently proven the global existence and dissipativity of strong solutions in
the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case
with small inertial coefficient and arbitrary growth rate of the nonlinearity.
The present contribution studies the long-time behavior of rather weak (energy)
solutions of that equation and it is a natural complement of the results of our
previous papers. Namely, we prove here that the attractors for energy and
strong solutions coincide for both the cases mentioned above. Thus, the energy
solutions are asymptotically smooth. In addition, we show that the non-smooth
part of any energy solution decays exponentially in time and deduce that the
(smooth) exponential attractor for the strong solutions constructed previously
is simultaneously the exponential attractor for the energy solutions as well
On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities
We study a non-local variant of a diffuse interface model proposed by
Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical
species acting as nutrient. The system consists of a Cahn--Hilliard equation
coupled to a reaction-diffusion equation. For non-degenerate mobilities and
smooth potentials, we derive well-posedness results, which are the non-local
analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015).
Furthermore, we establish existence of weak solutions for the case of
degenerate mobilities and singular potentials, which serves to confine the
order parameter to its physically relevant interval. Due to the non-local
nature of the equations, under additional assumptions continuous dependence on
initial data can also be shown.Comment: 28 page
On the principal bifurcation branch of a third order nonlinear long-wave equation
We study the principal bifurcation curve of a third order equation which
describes the nonlinear evolution of several systems with a long--wavelength
instability. We show that the main bifurcation branch can be derived from a
variational principle. This allows to obtain a close estimate of the complete
branch. In particular, when the bifurcation is subcritical, the large amplitude
stable branch can be found in a simple manner.Comment: 11 pages, 3 figure
Longtime behavior of nonlocal Cahn-Hilliard equations
Here we consider the nonlocal Cahn-Hilliard equation with constant mobility
in a bounded domain. We prove that the associated dynamical system has an
exponential attractor, provided that the potential is regular. In order to do
that a crucial step is showing the eventual boundedness of the order parameter
uniformly with respect to the initial datum. This is obtained through an
Alikakos-Moser type argument. We establish a similar result for the viscous
nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In
this case the validity of the so-called separation property is crucial. We also
discuss the convergence of a solution to a single stationary state. The
separation property in the nonviscous case is known to hold when the mobility
degenerates at the pure phases in a proper way and the potential is of
logarithmic type. Thus, the existence of an exponential attractor can be proven
in this case as well
Womenâs Experience of Group Prenatal Care
Group prenatal care (GPNC) is an innovative alternative to individual prenatal care. In this longitudinal study we used ethnographic methods to explore African American and Hispanic womenâs experiences of receiving GPNC in two urban clinics. Methods included individual, in-depth, semistructured interviews of women and group leaders in GPNC, participant observation of GPNC sessions, and medical record review. GPNC offered positive experiences and met many of the womenâs expressed preferences regarding prenatal care. Six themes were identified, which represented separate aspects of womenâs experiences: investment, collaborative venture, a social gathering, relationships with boundaries, learning in the group, and changing self. Taken together, the themes conveyed the overall experience of GPNC. Women were especially enthusiastic about learning in groups, about their relationships with group leaders, and about having their pregnancy-related changes and fears normalized. There were also important boundaries on relationships between participants, and some women wished for greater privacy during physical examinations
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
Diffuse-interface model for rapid phase transformations in nonequilibrium systems
A thermodynamic approach to rapid phase transformations within a diffuse
interface in a binary system is developed. Assuming an extended set of
independent thermodynamic variables formed by the union of the classic set of
slow variables and the space of fast variables, we introduce finiteness of the
heat and solute diffusive propagation at the finite speed of the interface
advancing. To describe the transformation within the diffuse interface, we use
the phase-field model which allows us to follow the steep but smooth change of
phases within the width of diffuse interface. The governing equations of the
phase-field model are derived for the hyperbolic model, model with memory, and
for a model of nonlinear evolution of transformation within the
diffuse-interface. The consistency of the model is proved by the condition of
positive entropy production and by the outcomes of the fluctuation-dissipation
theorem. A comparison with the existing sharp-interface and diffuse-interface
versions of the model is given.Comment: 15 pages, regular article submitted to Physical Review
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