American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Compact uniform attractors for dissipative non-autonomous lattice dynamical systems
  • CPAA Home
  • This Issue
  • Next Article
    $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation
December  2007, 6(4): 1075-1086. doi: 10.3934/cpaa.2007.6.1075

3D convective Cahn--Hilliard equation

Alp Eden 1, and  Varga K. Kalantarov 2,

1. 

Department of Mathematics, Bogaziçi University, TUBITAK Feza Gürsey Institute, for Basic Sciences, Istanbul, Turkey
2. 

Department of Mathematics, Koç University, Istanbul, Turkey

Received  December 2006 Revised  May 2007 Published  September 2007

We consider the initial boundary value problem for the 3D convective Cahn - Hilliard equation with periodic boundary conditions. This gives rise to a continuous dynamical system on $\dot L^2(\Omega)$. Absorbing balls in $\dot L^2(\Omega), \dot H_{per}^1(\Omega)$ and $\dot H_{per}^2(\Omega)$ are shown to exist. Combining with the compactness property of the solution semigroup we conclude the existence of the global attractor. Restricting the dynamical system on the absorbing ball in $\dot H_{per}^2(\Omega)$ and using the general framework in Eden et. all. [5] the existence of an exponential attractor is guaranteed. This approach also gives an explicit upper estimate of the dimension of the exponential attractor, albeit of the global attractor.
Keywords: attractor, exponential attractor, fractal dimension., convective Cahn - Hilliard equation, Global existence.
Mathematics Subject Classification: Primary: 35B41, 37L30; Secondary: 35K5.
Citation: Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075
[1]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[2]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[3]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[4]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[5]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[6]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
[7]

Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277
[8]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021006
[9]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289
[10]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[11]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[12]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[13]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287
[14]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[15]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020051
[16]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141
[17]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[18]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105
[19]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[20]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences