CPAA
On the asymptotic behavior of the Caginalp system with dynamic boundary conditions
Ciprian G. Gal M. Grasselli
Communications on Pure & Applied Analysis 2009, 8(2): 689-710 doi: 10.3934/cpaa.2009.8.689
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.
keywords: exponential attractors Laplace-Beltrami operator dynamic boundary conditions Phase-field equation global attractors Łojasiewicz-Simon inequality convergence to equilibrium
DCDS
Longtime behavior of nonlocal Cahn-Hilliard equations
Ciprian G. Gal Maurizio Grasselli
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 145-179 doi: 10.3934/dcds.2014.34.145
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.
keywords: convergence to single stationary states. regular and singular potentials uniform boundedness of solutions degenerate mobility nonlocal interactions well-posedness Cahn-Hilliard equation exponential attractors
CPAA
Robust exponential attractors for a conserved Cahn-Hilliard model with singularly perturbed boundary conditions
Ciprian G. Gal
Communications on Pure & Applied Analysis 2008, 7(4): 819-836 doi: 10.3934/cpaa.2008.7.819
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.
keywords: Cahn-Hilliard equations singular perturbations. dynamic boundary conditions exponential attractors global attractors
DCDS-S
Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions
Ciprian G. Gal Alain Miranville
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 113-147 doi: 10.3934/dcdss.2009.2.113
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.
keywords: global attractors Łojasiewicz-Simon inequality convergence to equilibrium. robust exponential attractors dynamic boundary conditions Viscous and non-viscous Cahn-Hilliard equations non-isothermal Cahn-Hilliard equations
DCDS-B
Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions
Ciprian G. Gal Maurizio Grasselli
Discrete & Continuous Dynamical Systems - B 2013, 18(6): 1581-1610 doi: 10.3934/dcdsb.2013.18.1581
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.
keywords: robust exponential attractors memory relaxation. dynamic boundary conditions Cahn-Hilliard equations global attractors
DCDS
Longtime behavior for a model of homogeneous incompressible two-phase flows
Ciprian G. Gal Maurizio Grasselli
Discrete & Continuous Dynamical Systems - A 2010, 28(1): 1-39 doi: 10.3934/dcds.2010.28.1
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.
keywords: incompressible fluids exponential attractors Navier-Stokes equations convergence to equilibria. Allen-Cahn equations nematic liquid crystals fractal dimension global attractors
EECT
Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions
Ciprian G. Gal Mahamadi Warma
Evolution Equations & Control Theory 2016, 5(1): 61-103 doi: 10.3934/eect.2016.5.61
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.
keywords: elliptic problem fractional Wentzell boundary conditions The fractional Laplace operator fractional Dirichlet-to-Neumann operator. exponential attractor global attractor fractional Steklov operator semilinear reaction-diffusion equation
DCDS
Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions
Ciprian G. Gal Mahamadi Warma
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1279-1319 doi: 10.3934/dcds.2016.36.1279
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.
keywords: global attractor asymptotic behavior. convergence to steady states The fractional Laplace operator semi-linear reaction-diffusion equation fractional Neumann and Robin boundary conditions on open sets
DCDS
The non-isothermal Allen-Cahn equation with dynamic boundary conditions
Ciprian G. Gal Maurizio Grasselli
Discrete & Continuous Dynamical Systems - A 2008, 22(4): 1009-1040 doi: 10.3934/dcds.2008.22.1009
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.
keywords: global attractors exponential attractors Phase-field systems dynamic boundary conditions
CPAA
Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows
Ciprian G. Gal T. Tachim Medjo
Communications on Pure & Applied Analysis 2014, 13(6): 2229-2252 doi: 10.3934/cpaa.2014.13.2229
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.
keywords: Lagrange averaged Navier-Stokes-$\alpha$ trajectory attractors. Allen-Cahn system phase transition

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