Displaying similar documents to “ Some models of Cahn-Hilliard equations in nonisotropic media ”

On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization

Ahmad Makki (2016)

Applications of Mathematics

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We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give...

Global Attractor for the Convective Cahn-Hilliard Equation

Xiaopeng Zhao, Changchun Liu (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is concerned with the convective Cahn-Hilliard equation. We use a classical theorem on existence of a global attractor to derive that the convective Cahn-Hilliard equation possesses a global attractor on some subset of H².

Attractors for general operators

Alain Miranville (2003)

Applications of Mathematics

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In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.

On the definition of strange nonchaotic attractor

Lluís Alsedà, Sara Costa (2009)

Fundamenta Mathematicae

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The aim of this paper is twofold. On the one hand, we want to discuss some methodological issues related to the notion of strange nonchaotic attractor. On the other hand, we want to formulate a precise definition of this kind of attractor, which is "observable" in the physical sense and, in the two-dimensional setting, includes the well known models proposed by Grebogi et al. and by Keller, and a wide range of other examples proposed in the literature. Furthermore, we analytically prove...

Global attractors for a tropical climate model

Pigong Han, Keke Lei, Chenggang Liu, Xuewen Wang (2023)

Applications of Mathematics

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This paper is devoted to the global attractors of the tropical climate model. We first establish the global well-posedness of the system. Then by studying the existence of bounded absorbing sets, the global attractor is constructed. The estimates of the Hausdorff dimension and of the fractal dimension of the global attractor are obtained in the end.

Unique global solvability of 1D Fried-Gurtin model

Zenon Kosowski (2007)

Applicationes Mathematicae

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We investigate a 1-dimensional simple version of the Fried-Gurtin 3-dimensional model of isothermal phase transitions in solids. The model uses an order parameter to study solid-solid phase transitions. The free energy density has the Landau-Ginzburg form and depends on a strain, an order parameter and its gradient. The problem considered here has the form of a coupled system of one-dimensional elasticity and a relaxation law for a scalar order parameter. Under some physically justified...

Existence and uniqueness of solutions for a three-dimensional thermoelastic system

Irena Pawłow, Antoni Żochowski

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A thermodynamically consistent model of shape memory alloys in three dimensions is studied. The thermoelastic system, based on the strain tensor, its gradient and the absolute temperature, is a generalization of the well-known one-dimensional Falk model. The key assumptions concerning the form of constitutive relations are discussed. The detailed and selfcontained proof of the global-in-time existence and uniqueness of solutions is presented.

Asymptotic behavior of a sixth-order Cahn-Hilliard system

Alain Miranville (2014)

Open Mathematics

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Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.

Mathematical analysis for the peridynamic nonlocal continuum theory

Qiang Du, Kun Zhou (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided.

Comparative Assessment of Nonlocal Continuum Solvent Models Exhibiting Overscreening

Baihua Ren, Jaydeep P. Bardhan (2017)

Molecular Based Mathematical Biology

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Nonlocal continua have been proposed to offer a more realistic model for the electrostatic response of solutions such as the electrolyte solvents prominent in biology and electrochemistry. In this work, we review three nonlocal models based on the Landau-Ginzburg framework which have been proposed but not directly compared previously, due to different expressions of the nonlocal constitutive relationship. To understand the relationships between these models and the underlying physical...

Global existence and uniqueness of weak solutions to Cahn-Hilliard-Gurtin system in elastic solids

Irena Pawłow, Wojciech M. Zajączkowski (2008)

Banach Center Publications

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In this paper we study the Cahn-Hilliard-Gurtin system describing the phase-separation process in elastic solids. The system has been derived by Gurtin (1996) as an extension of the classical Cahn-Hilliard equation. For a version with viscosity we prove the existence and uniqueness of a weak solution on an infinite time interval and derive an absorbing set estimate.

Remarks on the Fractal Dimension of Bi-Space Global and Exponential Attractors

Jan W. Cholewa, Radoslaw Czaja, Gianluca Mola (2008)

Bollettino dell'Unione Matematica Italiana

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Bi-space global and exponential attractors for the time continuous dynamical systems are considered and the bounds on their fractal dimension are discussed in the context of the smoothing properties of the system between appropriately chosen function spaces. The case when the system exhibits merely some partial smoothing properties is also considered and applications to the sample problems are given.

Multilevel Modeling of the Forest Resource Dynamics

I. N. Vladimirov, A. K. Chudnenko (2009)

Mathematical Modelling of Natural Phenomena

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We examine the theoretical and applications-specific issues relating to modeling the temporal and spatial dynamics of forest ecosystems, based on the principles of investigating dynamical models. When developing the predictive dynamical models of forest resources, there is a possibility of achieving uniqueness of the solutions to equations by taking into account the initial and boundary conditions of the solution, and the conditions of the geographical environment. We present the results...