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 Global attractor for a class of nonlinear generalized Kirchhoff models

2016 ◽  
Vol 12 (8) ◽  
pp. 6452-6462 ◽  
Author(s):  
Penghui Lv ◽  
Jingxin Lu ◽  
Guoguang Lin
Keyword(s):  
Boundary Value Problem ◽  
Global Attractor ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Natural Energy ◽  
Long Time ◽  
Initial Boundary Value

The paper studies the long time behavior of solutions to the initial boundary value problem(IBVP) for a class of Kirchhoff models flow  .We establish the well-posedness, theexistence of the global attractor in natural energy space

scholarly journals Long-time behavior of solutions for a fractional diffusion problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ailing Qi ◽  
Die Hu ◽  
Mingqi Xiang
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Attractors ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Initial Boundary Value

AbstractThis paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: $$ u_{t}+M\bigl([u]_{s,p}^{p}\bigr) (-\Delta )_{p}^{s}u+f(x,u)=g(x)\quad \text{in } \Omega \times (0, \infty ), $$ u t + M ( [ u ] s , p p ) ( − Δ ) p s u + f ( x , u ) = g ( x ) in  Ω × ( 0 , ∞ ) , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N is a bounded domain with Lipschitz boundary, $N>ps$ N > p s , $0< s<1<p$ 0 < s < 1 < p , $M:[0,\infty )\rightarrow [0,\infty )$ M : [ 0 , ∞ ) → [ 0 , ∞ ) is a nondecreasing continuous function, $[u]_{s,p}$ [ u ] s , p is the Gagliardo seminorm of u, $f:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ f : Ω × R → R and $g\in L^{2}(\Omega )$ g ∈ L 2 ( Ω ) . With general assumptions on f and g, we prove the existence of global attractors in proper spaces. Then, we show that the fractal dimensional of global attractors is infinite provided some conditions are satisfied.

scholarly journals Existence and long-time behavior of solutions to the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory

2021 ◽  
Author(s):  
Nguyen Toan
Keyword(s):  
Stokes Equations ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Time Behavior ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Voigt Model ◽  
Initial Boundary Value

In this paper, we study the long-time dynamical behavior of the non-autonomous velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory. We first investigate the existence and uniqueness of weak solutions to the initial boundary value problem for above-mentioned model. Next, we prove the existence of uniform attractor of this problem, where the time-dependent forcing term $f \in L^2_b(\mathbb{R}; H^{-1}(\Omega))$ is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in Yue, Wang (Comput. Math. Appl., 2020) in the case of non-autonomous and contain memory kernels which have not been studied before.

High order approximations in space and time of a sixth order Cahn–Hilliard equation

2015 ◽  
Vol 30 (6) ◽  
Author(s):  
Oleg Boyarkin ◽  
Ronald H. W. Hoppe ◽  
Christopher Linsenmann
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Sixth Order ◽  
Fourth Order Equation ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractWe consider an initial-boundary value problem for a sixth order Cahn-Hilliard equation describing the formation of microemulsions. Based on a Ciarlet-Raviart type mixed formulation as a system consisting of a second order and a fourth order equation, the spatial discretization is done by a C

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
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