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### Open Access Journals

CPAA

We consider a phase-field system of Caginalp type on a
three-dimensional bounded domain. The order parameter $\psi $
fulfills a dynamic boundary condition, while the (relative)
temperature $\theta $ is subject to a boundary condition of
Dirichlet, Neumann, Robin or Wentzell type. The corresponding
class of initial and boundary value problems has already been
studied by the authors, proving well-posedness results and the
existence of global as well as exponential attractors. Here we
intend to show first that the previous analysis can be redone for
larger phase-spaces, provided that the bulk potential has a
fourth-order growth at most whereas the boundary potential has an
arbitrary polynomial growth. Moreover, assuming the potentials to
be real analytic, we demonstrate that each
trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon type inequality. We also obtain a convergence
rate estimate.

DCDS

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.

CPAA

In this article, we construct a robust (that is, lower and upper
semi-continuous) family of exponential attractors for a conserved
Cahn-Hilliard model with the perturbation parameter in the
boundary conditions. We note that the existence of a global
attractor with finite dimension follows. Moreover, we prove the
upper semi-continuity of the limiting attractor with respect to
the family of perturbed global attractors.

DCDS-S

We consider a model of non-isothermal phase
separation taking place in a confined container. The order
parameter $\phi $ is governed by a viscous or non-viscous
Cahn-Hilliard type equation which is coupled with a heat equation
for the temperature $\theta $. The former is subject to a
nonlinear dynamic boundary condition recently proposed by
physicists to account for interactions with the walls, while the
latter is endowed with a standard (Dirichlet, Neumann or Robin)
boundary condition. We indicate by $\alpha $ the viscosity
coefficient, by $\varepsilon $ a (small) relaxation parameter
multiplying $\partial _{t}\theta $ in the heat equation and by
$\delta $ a small latent heat coefficient (satisfying $\delta \leq
\lambda \alpha $, $\delta \leq \overline{\lambda }\varepsilon $, $\lambda ,
\overline{\lambda }>0$) multiplying $\Delta \theta $ in the
Cahn-Hilliard equation and $\partial _{t}\phi $ in the heat
equation. Then, we construct a family of exponential attractors
$\mathcal{M}_{\varepsilon ,\delta ,\alpha }$
which is a robust perturbation of an exponential attractor $\mathcal{M}
_{0,0,\alpha }$ of the (isothermal) viscous ($\alpha >0$)
Cahn-Hilliard
equation, namely, the symmetric Hausdorff distance between $\mathcal{M}
_{\varepsilon ,\delta ,\alpha }$ and $\mathcal{M}_{0,0,\alpha }$
goes to 0, for each fixed value of $\alpha >0,$ as $(
\varepsilon ,\delta) $ goes to $(0,0),$ in an explicitly
controlled way. Moreover, the robustness of this family of
exponential attractors $\mathcal{M}_{\varepsilon ,\delta ,\alpha
}$ with respect to $( \delta ,\alpha ) \rightarrow
( 0,0) ,$ for each fixed value of $\varepsilon >0,$ is
also obtained. Finally, assuming that the nonlinearities are real
analytic, with no growth restrictions, the convergence of
solutions to single equilibria, as time goes to infinity, is also
proved.

DCDS-B

We consider a modified Cahn-Hiliard equation
where the velocity of the order parameter $u$ depends on the past
history of $\Delta
\mu $, $\mu $ being the chemical potential with an additional viscous term $
\alpha u_{t},$ $\alpha >0.$ In addition, the usual no-flux
boundary condition for $u$ is replaced by a nonlinear dynamic
boundary condition which accounts for possible interactions with
the boundary. The aim of this work is to analyze the passage to
the singular limit when the memory kernel collapses into a Dirac
mass. In particular, we discuss the convergence of solutions on
finite time-intervals and we also establish stability results for
global and exponential attractors.

DCDS

We consider a diffuse interface model for the evolution of an
iso-thermal incompressible two-phase flow in a two-dimensional bounded
domain. The model consists of the Navier-Stokes equation for the fluid
velocity

**u**coupled with a convective Allen-Cahn equation for the order (phase) parameter $\phi$, both endowed with suitable boundary conditions. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase space which possesses the global attractor $ \mathcal{A}$. Then we establish the existence of an exponential attractor $ \mathcal{E}$ which entails that $\mathcal{A}$ has finite fractal dimension. This dimension is then estimated in terms of some model parameters. Moreover, assuming the potential to be real analytic, we demonstrate that, in absence of external forces, each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon inequality. We also obtain a convergence rate estimate. Finally, we discuss the case where $\phi $ is forced to take values in a bounded interval, e.g., by a so-called singular potential.
EECT

We investigate a class of semilinear parabolic and elliptic problems with
fractional dynamic boundary conditions. We introduce two new operators, the
so-called fractional Wentzell Laplacian and the fractional Steklov operator,
which become essential in our study of these nonlinear problems. Besides
giving a complete characterization of well-posedness and regularity of
bounded solutions, we also establish the existence of finite-dimensional
global attractors and also derive basic conditions for blow-up.

DCDS

We investigate the long term behavior in terms of finite dimensional global
attractors and (global) asymptotic stabilization to steady states, as time
goes to infinity, of solutions to a non-local semilinear reaction-diffusion
equation associated with the fractional Laplace operator on non-smooth
domains subject to Dirichlet, fractional Neumann and Robin boundary
conditions.

DCDS

We consider a model of nonisothermal phase transitions taking
place in a bounded spatial region. The order parameter $\psi$ is governed by
an Allen-Cahn type equation which is coupled with the equation for the
temperature $\theta$. The former is subject to a dynamic boundary condition
recently proposed by some physicists to account for interactions with the
walls. The latter is endowed with a boundary condition which can be a
standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell
type. We thus formulate a class of initial and boundary value problems whose
local existence and uniqueness is proven by means of a fixed point argument.
The local solution becomes global owing to suitable a priori estimates. Then
we analyze the asymptotic behavior of the solutions within the theory of
infinite-dimensional dynamical systems. In particular, we demonstrate the
existence of the global attractor as well as of an exponential attractor.

CPAA

In this article we study the relations between the long-time
dynamics of the 3D Allen-Cahn-LANS-$\alpha$ model and the exact 3D
Allen-Cahn-Navier-Stokes system. Following the idea of [26],
we prove that bounded set of solutions of the
Allen-Cahn-LANS-$\alpha$ model converge to the trajectory attractor
$\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes system as time goes to $+
\infty $ and $\alpha$ approaches $0^+. $ In particular we show that
the trajectory attractors $\mathcal{U}_{\alpha} $ of the 3D
Allen-Cahn-LANS-$\alpha$ model converges to the trajectory attractor
$\mathcal{U}_0 $ of the 3D Allen-Cahn-Navier-Stokes as $ \alpha $ approaches
$0^+. $ Let us mention that the strong nonlinearity that results
from the coupling of the convective Allen-Cahn system and the
LANS-$\alpha$ equations makes the analysis of the problem considered
in this article more involved.

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