scholarly journals A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term

2011 ◽  
Keyword(s):  
Discrete Approximation ◽  
Inertial Term ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Long Time ◽  
Fully Discrete Approximation ◽  
Cahn Hilliard Equation

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

scholarly journals Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility

2020 ◽  
Vol 32 (1) ◽  
pp. 89-112
Author(s):  
BENOÎT PERTHAME ◽  
ALEXANDRE POULAIN
Keyword(s):  
Hilliard Equation ◽  
Degenerate Mobility ◽  
Long Time ◽  
Single Well ◽  
Cahn Hilliard Equation ◽  
Long Time Asymptotics ◽  
Living Tissues ◽  
Entropy Structure ◽  
Second Order Equations ◽  
Relaxed System

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

scholarly journals On the 3D Cahn–Hilliard equation with inertial term

2009 ◽  
Vol 9 (2) ◽  
pp. 371-404 ◽  
Author(s):  
Maurizio Grasselli ◽  
Giulio Schimperna ◽  
Antonio Segatti ◽  
Sergey Zelik
Keyword(s):  
Inertial Term ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

scholarly journals Analysis of a Dynamic Viscoelastic Contact Problem with Normal Compliance, Normal Damped Response, and Nonmonotone Slip Rate Dependent Friction

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Mikaël Barboteu ◽  
David Danan
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Slip Rate ◽  
Approximation Schemes ◽  
Discrete Approximation ◽  
Normal Compliance ◽  
Fully Discrete ◽  
Rate Dependent ◽  
Fully Discrete Approximation ◽  
Normal Damped Response

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.

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