The viscous Cahn-Hilliard equation with inertial term

We consider a hyperbolic relaxation of the viscous Cahn-Hilliard equation. This equation describes the early stages of spinodal decomposition in certain glasses. We establish the existence of families of exponential attractors and inertial manifolds which are continuous at any parameter of viscosity ?≥0. Continuity properties of the global attractors are also examined.     

The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations

We study the structure of diffusive layers in solutions of unstable nonlinear diffusion equations. These equations are regularizations of the forward-backward heat equation and have diffusion coefficients that become negative. Such models include the Cahn-Hilliard equation and the pseudoparabolic viscous diffusion equation. Using singular perturbation methods we show that the balance between diffusion and higher-order regularization terms uniquely determines the interface structure in these equations. It is shown that the well-known “equal area” rule for the Cahn-Hilliard equation is a special case of a more general rule for shock construction in the viscous Cahn-Hilliard equation.     

On the convergence of a fourth order evolution equation to the Allen-Cahn equation

We construct special sequences of solutions to a fourth order nonlinear parabolic equation of Cahn-Hilliard/Allen-Cahn type, converging to the second order Allen-Cahn equation. We consider the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. The proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with “good signs”.     

Long Time Behavior of the Cahn-Hilliard Equation with Irregular Potentials and Dynamic Boundary Conditions

The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered.The existence of the global attractor is proved and the long time behavior of the trajectories,namely,the convergence to steady states,is studied.     

Higher-order models in phase separation

Our aim in this paper is to study higher-order (in space) Allen-Cahn and Cahn-Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor.     

A conservative nonlinear difference scheme for the viscous Cahn-Hilliard equation

Numerical solutions for the viscous Cahn-Hilliard equation are considered using the crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution inH ¹-norm are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13].     

Generating pairs for the sporadic groupRu

Analytical solutions for the viscous Cahn-Hilliard equation are considered. Existence and uniqueness of the solution are shown. The exponential decay of the solution inH ²-norm, which is an improvement of the result in Elliott and Zheng[5]. We also compare the early stages of evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation, which has been given as an open question in Novick-Cohen[8].     

The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in L_p-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.     

Maximal attractor for an Ostwald ripening model

We consider a family of phase-field models that couples a Cahn-Hilliard with several Allen-Cahn equations. We show that such family of phase-field models possesses a maximal attractor with finite fractal dimension.     

Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=u^m(1-u)^m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.     

Dynamics of the viscous Cahn-Hilliard equation

We study generalized viscous Cahn-Hilliard problems with nonlinearities satisfying critical growth conditions in , where Ω is a bounded smooth domain in Rn, n?3. In the critical growth case, we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical. For p=2 and almost critical dissipative nonlinearities we prove global well posedness, existence of global attractors in and, uniformly with respect to the viscosity parameter, L^∞(Ω) bounds for the attractors. Finally, we obtain a result on continuity of regular attractors which shows that, if n=3,4, the attractor of the Cahn-Hilliard problem coincides (in a sense to be specified) with the attractor for the corresponding semilinear heat equation.     

On the variational formulation and implementation of Allen-Cahn and Cahn-Hilliard-type phase field theories

Phase field theory is a promising framework for analyzing evolving microstructures in materials. Phenomena like those related to microstructures in Ni-based superalloys, twin structures in martensites or precipitation in Al-alloys can be predicted by phase field theory. While phase transformations such as those characterizing twinning are captured by an Allen-Cahn-type approach, a Cahn-Hilliard-type formulation is used, if the respective interface motion is driven by the concentration of the species. Although the Allen-Cahn and the Cahn-Hilliard formulation are indeed different, they do share some similarities. To be more precise, a Cahn-Hilliard model is obtained by enforcing balance of mass in the Allen-Cahn approach. Within an energy-based formulation this can be implemented by adding additional energy terms to the underlying Allen-Cahn energy. Such a universal energy-based framework is elaborated in this presentation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global attractors for the viscous hyperelastic‐rod wave equation

We consider the viscous hyperelastic‐rod wave equation subject to an external force, where the viscous term is given by second order differential operator in divergence form. Under some mild assumptions on the viscous term, first, we establish the global well‐posedness in both the periodic case and the case of the whole line, afterwards, we show the existence of global attractors for the two cases, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential attractors for a class of evolution equations by a decompoition method. II. The non-autonomous case

Our aim in this Note is to extend to the non-autonomous case the result of the note [8] in which we proved the existence of finite-dimensional attractors for a general class of equations containing some models of Cahn-Hilliard equations introduced in [6].     

Global attractors for the initial value problems of Cohn-Hilliard equations on compact manifolds

In this paper we establish some theorems which are concerned with the equivalent norms of Sobolev spaces on a Riemannian manifold. Using the theorems we prove the existence of global attractors for the initial value problem of Cahn-Hilliard equations. The estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors are also obtained.     

Stability of the Semi-Implicit Method for the Cahn-Hilliard Equation with Logarithmic Potentials

We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme, which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semiimplicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.     

Optimal control problem for viscous Cahn-Hilliard equation

This paper is concerned with the viscous Cahn-Hilliard equation, which arises in the dynamics of viscous first order phase transitions in cooling binary solutions. The optimal control under boundary condition is given and the existence of optimal solution to the equation is proved.     

On the hyperbolic relaxation of the one-dimensional Cahn-Hilliard equation

We consider the one-dimensional Cahn-Hilliard equation with an inertial term ?utt, for ??0. This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S?(t) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors {M?}, whose common basins of attraction are the whole phase-space.     

3‐D viscous Cahn–Hilliard equation with memory

We deal with the memory relaxation of the viscous Cahn–Hilliard equation in 3‐D, covering the well‐known hyperbolic version of the model. We study the long‐term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient δ. In particular we construct a family of exponential attractors, which is robust as both ε and δ go to zero, provided that ε is linearly controlled by δ. Copyright © 2008 John Wiley & Sons, Ltd.