scholarly journals Long-time behavior of the one-phase Stefan problem in periodic and random media

2018 ◽  
Vol 11 (5) ◽  
pp. 991-1010
Author(s):  
Norbert Požár ◽  
◽  
Giang Thi Thu Vu ◽  
Keyword(s):  
Stefan Problem ◽  
Random Media ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
The One

scholarly journals DISPERSIVE EFFECTIVE MODELS FOR WAVES IN HETEROGENEOUS MEDIA

2011 ◽  
Vol 21 (09) ◽  
pp. 1871-1899 ◽  
Author(s):  
AGNES LAMACZ
Keyword(s):  
Heterogeneous Media ◽  
Variable Coefficients ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Smooth Functions ◽  
Long Time ◽  
Constant Coefficients ◽  
The One ◽  
Effective Models ◽  
Dispersive Models

We study the long-time behavior of waves in a strongly heterogeneous medium, starting from the one-dimensional scalar wave equation with variable coefficients. We assume that the coefficients are periodic with period ε and ε > 0 is a small length parameter. Our main result concerns homogenization and consists in the rigorous derivation of two different dispersive models. The first is a fourth-order equation with constant coefficients including powers of ε. In the second model, the ε-dependence is completely avoided by considering a third-order linearized Korteweg–de Vries equation. Our result is that both simplified models describe the long-time behavior well. An essential tool in our analysis is an adaption operator which modifies smooth functions according to the periodic structure of the medium.

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 ) .

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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