Long-time behavior of the Cahn–Hilliard equation with dynamic boundary condition

2020 ◽  
Vol 6 (1) ◽  
pp. 283-309 ◽  
Author(s):  
Alain Miranville ◽  
Hao Wu
Keyword(s):  
Boundary Condition ◽  
Time Behavior ◽  
Dynamic Boundary Condition ◽  
Long Time Behavior ◽  
Hilliard Equation ◽  
Dynamic Boundary ◽  
Long Time ◽  
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 A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Makoto Okumura ◽  
Takeshi Fukao ◽  
Daisuke Furihata ◽  
Shuji Yoshikawa
Keyword(s):  
Boundary Condition ◽  
Numerical Solutions ◽  
Normal Derivative ◽  
Second Order ◽  
Dynamic Boundary Condition ◽  
Central Difference ◽  
Structure Preserving ◽  
Hilliard Equation ◽  
Dynamic Boundary ◽  
Cahn Hilliard Equation

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>

scholarly journals Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction

2018 ◽  
Vol 38 (8) ◽  
pp. 3765-3788 ◽  
Author(s):  
Annalisa Iuorio ◽  
◽  
Stefano Melchionna ◽  
Keyword(s):  
Time Behavior ◽  
Long Time Behavior ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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