Non-blow-up phenomenon for the Cahn-Hilliard equation with non-constant mobility

In this paper we prove that the finite time blow-up phenomenon might not occur for the Cahn-Hilliard equation with non-constant mobility and cubic nonlinearity, which is quite different from the case of constant mobility. We reveal such a phenomenon under some structure condition on the mobility.     

Finite Speed of Propagation of Perturbations for the Cahn-Hilliard Equation with Degenerate Mobility

This paper is devoted to the Cabn-Hilliard equation with degenerate mobility in two spatial variables with a typical case modelling thin viscous film spreading over a solid surface. We establish tbe existence of radial symmetric solutions with the property of finite speed of perturbations.     

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 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.     

一类非局部Cahn-Hilliard方程弱解的存在唯一性

研究一类对流非局部Cahn-Hilliard方程的Neumann问题.通过一致Schauder估计和Leray-Schauder不动点定理,得到了该问题经典解的存在唯一性.进而,利用弱收敛方法得到了该问题弱解的存在唯一性.     

Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions

This paper is concerned with the asymptotic behavior of solution to the Cahn-Hilliard equation

(0.1)     

A multigrid method for the Cahn-Hilliard equation with obstacle potential

We present a multigrid finite element method for the deep quench obstacle Cahn-Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ. Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn-Hilliard equation.     

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.     

The Cahn-Hilliard type equations with periodic potentials and sources

We discuss the existence, uniqueness and asymptotic estimates of solutions to the Cahn-Hilliard type equations with time periodic potentials and sources.     

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

Two Dimensional Cahn-Hilliard Equation with Concentration Dependent Mobility and Gradient Dependent Potential

In this paper we consider the initial boundary value problem of Cahn-Hilliard equation with concentration dependent mobility and gradient dependent potential. By the L~P type estimates and the theory of Morrey spaces,we prove the Holder continuity of the solutions.Then we obtain the existence of global classical solutions.The present work can be viewed as an extension to the previous work on the Cahn-Hilliard equation with concentration dependent mobility and potential.     

具浓度相关迁移率和梯度相关位势的3维Cahn-Hilliard方程

In this paper we investigate the initial boundary value problem of Cahn-Hilliard equation with concentration dependent mobility and gradient dependent potential.By the energy method and the theory of Campanato spaces,we prove the existence and the uniqueness of classical solutions in 3-dimensional space.     

A class of degenerate parabolic equations with a concentrated nonlinear source

In this paper, we consider the Cauchy problem for a class of degenerate parabolic equations with a concentrated nonlinear source. We obtain the existence of the generalized solutions for the problem based on some a priori estimates on solutions.     

Cahn-Hilliard方程的动态分歧

利用线性全连续场的谱理论,中心流形约化方法与非线性耗散系统吸引子分歧理论,研究了Cahn-Hilliard方程的动态分歧,给出了发生分歧的条件及临界点,并给出了在Neumann边界条件下,方程分歧出的稳定奇点吸引子和鞍点的表达式.     

A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation

This paper is concerned with the optimal temporal decay estimates on the solutions of the Cauchy problem of the Cahn-Hilliard equation. It is shown in Liu, Wang and Zhao (2007) [11] that such a Cauchy problem admits a unique global smooth solution u(t,x) provided that the smooth nonlinear function φ(u) satisfies a local growth condition. Furthermore if φ(u) satisfies a somewhat stronger local growth condition, the optimal temporal decay estimates on u(t,x) are also obtained in Liu, Wang and Zhao (2007) [11]. Thus a natural question is how to deduce the optimal temporal decay estimates on u(t,x) only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding Cauchy problem and the main purpose of this paper is devoted to this problem. Our analysis is motivated by the technique developed recently in Ukai, Yang and Zhao (2006) [15] with a slight modification.     

On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions

We study the global existence of solutions of the nonlinear degenerate wave equation (ρ?0)

     

A Note on the Viscous Cahn-Hilliard Equation

In this note, we study the global existence of classical solutions for the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5. Based on the Schauder type estimates and energy estimates, we establish the global existence of classical solutions.     

具浓度相关迁移率的对流扩散Cahn-Hilliard方程

In this paper, we study the global existence of classical solutions for the convective-diffusive Cahn-Hilliard equation with concentration dependent mobility. Based on the Schauder type estimates, we establish the global existence of classicalsolutions.     

On a degenerate parabolic equation arising in pricing of Asian options

We study a certain one-dimensional, degenerate parabolic partial differential equation with a boundary condition which arises in pricing of Asian options. Due to degeneracy of the partial differential operator and the non-smooth boundary condition, regularity of the generalized solution of such a problem remained unclear. We prove that the generalized solution of the problem is indeed a classical solution.     

Degenerate nonlocal Cahn-Hilliard equations: Well-posedness,regularity and local asymptotics

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.