Optimal control of the convective Cahn–Hilliard equation

This article studies the problem for optimal control of the convective Cahn–Hilliard equation in one-space dimension. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.     

A numerical analysis of the Cahn–Hilliard equation with non-permeable walls

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.     

A Fourier spectral method for fractional-in-space Cahn–Hilliard equation

In this paper, a fractional extension of the Cahn–Hilliard (CH) phase field model is proposed, i.e. the fractional-in-space CH equation. The fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case. An unconditionally energy stable Fourier spectral scheme is developed to solve the fractional equation with periodic or Neumann boundary conditions. This method is of spectral accuracy in space and of second-order accuracy in time. The main advantages of this method are that it yields high precision and high efficiency. Moreover, an extra stabilizing term is added to obey the energy decay property while maintaining accuracy and simplicity. Numerical experiments are presented to confirm the accuracy and effectiveness of the proposed method.     

The Cahn–Hilliard equation with time-dependent constraint

We propose an abstract variational inequality formulation of the Cahn–Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems.     

On the 3D Cahn–Hilliard equation with inertial term

We study the modified Cahn–Hilliard equation proposed by Galenko et al. in order to account for rapid spinodal decomposition in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration multiplied by a (small) positive coefficient . Thus, in absence of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize in finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions, the problem of finding a unique solution satisfying given initial and boundary conditions is far from being trivial. A fairly complete analysis of the 2D case has been recently carried out by Grasselli, Schimperna and Zelik. The 3D case is still rather poorly understood but for the existence of energy bounded solutions. Taking advantage of this fact, Segatti has investigated the asymptotic behavior of a generalized dynamical system which can be associated with the equation. Here we take a step further by establishing the existence and uniqueness of a global weak solution, provided that is small enough. More precisely, we show that there exists such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of which goes to + ∞ as tends to 0. This result allows us to construct a semigroup on an appropriate (bounded) phase space and, besides, to prove the existence of a global attractor. Finally, we show a regularity result for the attractor by using a decomposition method and we discuss the existence of an exponential attractor.      

Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn–Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn–Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.     

Singularly perturbed 1D Cahn–Hilliard equation revisited

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors ${\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M_e}, e 3 0{\{{\mathcal M}_\epsilon\}, \epsilon\geq 0} being the perturbation parameter, such that the map e? M_e{\epsilon \mapsto {\mathcal M}_\epsilon} is H?lder continuous. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e.{\epsilon.} Continuity properties of global attractors and inertial manifolds are also examined.     

Cahn–Hilliard equation with nonlocal singular free energies

Annali di Matematica Pura ed Applicata (1923 -) - We consider a Cahn–Hilliard equation which is the conserved gradient flow of a nonlocal total free energy functional. This functional is...     

The least squares spectral element method for the Cahn–Hilliard equation

The problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn–Hilliard (C–H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C–H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the Cahn–Hilliard equation. The LS-SEM is combined with a time–space coupled formulation and a high order continuity approximation by employing C¹¹p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn–Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C–H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.     

Stochastic Cahn–Hilliard equation with singular nonlinearity and reflection

We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 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 and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semigroup. We obtain existence and uniqueness of a solution for nonnegative initial conditions, results on the invariant measures, and on the reflection measures.     

Global attractor for the Cahn–Hilliard/Allen–Cahn system

We study the coupled Cahn–Hilliard/Allen–Cahn problem with constraints, which describes the isothermal diffusion-driven phase transition phenomena in binary systems. Our aim is to show the existence–uniqueness result and to construct the global attractor for the related dynamical system.     

A probabilistic approach to the Yang–Mills heat equation

We construct a parallel transport U in a vector bundle E, along the paths of a Brownian motion in the underlying manifold, with respect to a time dependent covariant derivative ∇ on E, and consider the covariant derivative ∇₀U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U⁻¹∇₀U of this covariant derivative has quadratic variation twice the Yang–Mills energy density (i.e., the square norm of the curvature 2-form) integrated along the Brownian motion, and that the drift of such processes vanishes if and only if ∇ solves the Yang–Mills heat equation. A monotonicity property for the quadratic variation of U⁻¹∇₀U is given, both in terms of change of time and in terms of scaling of U⁻¹∇₀U. This allows us to find a priori energy bounds for solutions to the Yang–Mills heat equation, as well as criteria for non-explosion given in terms of this quadratic variation.     

Cahn–Hilliard equation with two spatial variables. Pattern formation

Theoretical and Mathematical Physics - We consider the Cahn–Hilliard equation in the case where its solution depends on two spatial variables, with homogeneous Dirichlet and Neumann boundary...     

Complicated asymptotic behavior of solutions for the Cauchy problem of Cahn–Hilliard equation

In this paper, we study the Cauchy problem of the Cahn–Hilliard equation, and first reveal that the complicated asymptotic behavior of solutions can happen in high-order parabolic equation.