Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation

Heida, Martin

doi:10.1007/s10492-015-0085-7

Cited by 16 publications

(8 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From then on, many generalisations of the equation were considered and several applications to different areas of mathematics were explored; some examples are [19,20,21,23] and [22,12,13] for the degenerate version of the equation. See also [31,34] where the gradient flow structure of the problem is exploited, [9,30] for relations with geometric flows, and [46] for a stochastic version. It is also worthwhile mentioning the articles [45,9,43] where an asymptotic analysis is performed for the parameter ε → 0; the latter does so for the equation on an evolving surface.…”

Section: Background and Motivationmentioning

confidence: 99%

Cahn–Hilliard equations on an evolving surface

Caetano¹,

Elliott²

2021

Eur. J. Appl. Math

View full text Add to dashboard Cite

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

show abstract

Section: Background and Motivationmentioning

confidence: 99%

Cahn–Hilliard equations on an evolving surface

Caetano¹,

Elliott²

2021

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

“…Since then rigorous mathematical analysis for the Cahn-Hilliard system and its variants subject to the same dynamic boundary condition (1.6) have been performed. We refer to [11,19,21,35,50,74] for well-posedness results via different approaches, to [10,36,37,41,65,66] for regularity properties and long-time behavior in terms of global or exponential attractors, and to [11,18,38,42,88] for global asymptotic stability of single trajectories as time goes to infinity. Next, for the Cahn-Hilliard system subject to the second type of dynamic boundary conditions (1.8)-(1.9), well-posedness and long-time behavior was first investigated in [45] (see also [15]).…”

Section: Introductionmentioning

confidence: 99%

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

2019

Arch Rational Mech Anal

101

117

View full text Add to dashboard Cite

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn-Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints such as conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as t → +∞ and characterize the stability of local energy minimizers for the system.

show abstract

“…There are two famous phase‐field models for the evolution of field variables: the Cahn‐Hilliard nonlinear diffusion equation 7 used to describe the evolution of conserved field variables such as atomic concentration and the time‐dependent Ginzburg–Landau (Allen–Cahn) equation 8,9 used of representing the variation of the nonconserved field variables, for instance, orientation field in martensitic transformation. A large number of achievements have been made in theory 10‐14 and numerical studies 1‐6,15 of those two models.…”

Section: Introductionmentioning

confidence: 99%

Analysis for a new phase‐field model with periodic boundary conditions

Zhao

Mai

Cheng

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

This article is concerned with an initial‐boundary value problem (IBVP) for a new phase‐field model describing the evolution of structural phase transition in elastically deformable solid materials. The model consists of an elliptic‐parabolic system in which the displacement field and the order parameter both satisfy periodic boundary conditions. We prove the existence of global solutions to this IBVP by applying the method of continuation of local solutions and perform numerical simulations to investigate the microstructure evolution of MnNi alloys by using this new phase‐field model.

show abstract

Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation

Cited by 16 publications

References 49 publications

Cahn–Hilliard equations on an evolving surface

Cahn–Hilliard equations on an evolving surface

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

Analysis for a new phase‐field model with periodic boundary conditions

Contact Info

Product

Resources

About