On anisotropic caginalp phase-field type models with singular nonlinear terms

Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.     

On higher-order anisotropic Caginalp phase-fi eld systems with polynomial nonlinear terms

Our aim in this paper is to study higher-order (in space) anisotropic Caginalp phase-field systems. In particular, we obtain well-posedness results, as well as the existence of the global attractor and exponential attractor.     

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.     

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

In one space dimension, a non-local elastic model is based ona single integral law, giving the stress when the strain isknown at all spatial points. In this study, we first derivea higher-order Boussinesq equation using locally non-lineartheory of 1D non-local elasticity and then we are able to showthat under certain conditions the Cauchy problem is globallywell-posed.     

Nonlinear evolution inclusions arising from phase change models

The paper is devoted to the analysis of an abstract evolution inclusion with a non-invertible operator, motivated by problems arising in nonlocal phase separation modeling. Existence, uniqueness, and long-time behaviour of the solution to the related Cauchy problem are discussed in detail. The Italian authors would like to point out financial support from theMIUR-COFIN 2002 research program on “Free boundary problems in applied sciences”. The second author was supported by GA ČR under Grant No. 201/02/1058, and by GNAMPA of INDAM during his stay at Pavia in May 2003: in this respect, the kind hospitality of the Department of Mathematics in Pavia is gratefully acknowledged as well. The work also benefited from partial support of the IMATI of CNR in Pavia, Italy.     

Asymptotic behaviour of a generalized Cahn–Hilliard equation with a proliferation term

In this article, we are interested in the study of the asymptotic behaviour, in terms of finite-dimensional attractors, of a generalization of the Cahn–Hilliard equation with a proliferation term. Such a model has, in particular, applications in biology.     

Global Well-posedness and Global Attractor for Two-dimensional Zakharov-Kuznetsov Equation

The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in H^s(R²) for all 5/7 < s < 1 via using I-method in the context of atomic spaces. By means of the increment of modified energy, the existence of global attractor for the weakly damped, forced Zakharov-Kuznetsov equation is also established in H^s(R²) for 10/11 < s < 1.     

On phase separation points for one-dimensional models

In the paper, the one-dimensional model with nearest-neighbor interactions I _n , n ∈ Z, and the spin values ±1 is considered. It is known that, under some conditions on parameters of I _n , a phase transition occurs for this model. We define the notion of a phase separation point between two phases. We prove that the expectation value of this point is zero and its mean-square fluctuation is bounded by a constant C(β) which tends to ¼ as β → ∞, where β = 1/T and T is the temperature.     

一类非线性高阶Kirchhoff型方程的初边值问题

本文研究了一类具有非线性耗散项的高阶Kirchhoff型方程的初边值问题.通过构造稳定集讨论了此问题整体解的存在性,应用Nakao的差分不等式建立了解能量的衰减估计.在初始能量为正的条件下,证明了解在有限时间内发生blow-up,并且给出了解的生命区间估计.     

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size and the time step size . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

     

Effect of Space Dimensions on Equilibrium Solutions of Cahn-Hilliard and Conservative Allen-Cahn Equations

In this study, we investigate the effect of space dimensions on the equilibrium solutions of the Cahn-Hilliard (CH) and conservative Allen-Cahn (CAC) equations in one, two, and three dimensions. The CH and CAC equations are fourth-order parabolic partial and second-order integro-partial differential equations, respectively. The former is used to model phase separation in binary mixtures, and the latter is used to model mean curvature flow with conserved mass. Both equations have been used for modeling various interface problems. To study the space-dimension effect on both the equations, we consider the equilibrium solution profiles for symmetric, radially symmetric, and spherically symmetric drop shapes. We highlight the different dynamics obtained from the CH and CAC equations. In particular, we find that there is a large difference between the solutions obtained from these equations in three-dimensional space.     

Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.     

Unbounded One-Dimensional Global Attractor for the Damped Sine-Gordon Equation

Summary. The dynamical behavior of the damped sine-Gordon equation with homogeneous Neumann boundary condition is studied. It is shown that the equation has an unbounded one-dimensional global attractor in a suitable functional space when the ``damping' and the ``diffusing' are not very small. Received March 16, 1999; accepted December 10, 1999     

On a nonlocal phase separation model

An alternative to the Cahn-Hilliard model of phase separation for two-phase systems in a simplified isothermal case is given. The model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations. Using the free energy as Lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle.     

锥似凸映射下集值优化Henig有效解的高阶最优性条件(英文)

This paper deals with higher-order optimality conditions for Henig effcient solutions of set-valued optimization problems.By virtue of the higher-order tangent sets, necessary and suffcient conditions are obtained for Henig effcient solutions of set-valued optimization problems whose constraint condition is determined by a fixed set.     

一个具有Hamilton结构的反应扩散系统的空间离散化整体吸引子（英）

In recent years there has been a growing interest on discrete models, see e.g.[1].[2].We consider a reaction diffusion equation which space-independent system is a Hamilton system with one degree of freedom:     

On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces

We establish local and global well-posedness of the 2D dissipative quasi-geostrophic equation in critical mixed norm Lebesgue spaces. The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivial due to the nonlocal structure of the equation. Our approach is based on Kato's method using Picard's iteration, which can be adapted to the multi-dimensional case and other nonlinear non-local equations. We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.     

Phase-Field Modeling and Numerical Simulation for Ice Melting

In this paper, we propose a mathematical model and present numerical simulations for ice melting phenomena. The model is based on the phase-field modeling for the crystal growth. To model ice melting, we ignore anisotropy in the crystal growth model and introduce a new melting term. The numerical solution algorithm is a hybrid method which uses both the analytic and numerical solutions. We perform various computational experiments. The computational results confirm the accuracy and efficiency of the proposed method for ice melting.     

带有脉冲免疫和传染年龄的传染病模型的全局吸引性

讨论了带有脉冲免疫和传染年龄的传染病模型.传染类的恢复率是传染年龄的函数,当染病再生数小于1时,文章得到无病周期解是全局吸引的.如果总人口规模变化,也可得到类似的结论.最后,提出了带有脉冲免疫和传染年龄传染病模型待解决的问题.     

Global attractor in competitive Lotka-Volterra systems with retardation

Global attractivity is studied for a class of competitive Lotka-Volterra differential systems with retardation. Sufficient conditions, which contain a number of existing results as special instances, are provided for a system to have a single-point global attractor. By these conditions, predictions can be made either for coexistence and stability of all the species or for balance of survival and extinction.