Some models of Cahn-Hilliard equations in nonisotropic media

Alain Miranville

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 3, page 539-554
  • ISSN: 0764-583X

Abstract

top
We derive in this article some models of Cahn-Hilliard equations in nonisotropic media. These models, based on constitutive equations introduced by Gurtin in [19], take the work of internal microforces and also the deformations of the material into account. We then study the existence and uniqueness of solutions and obtain the existence of finite dimensional attractors.

How to cite

top

Miranville, Alain. " Some models of Cahn-Hilliard equations in nonisotropic media ." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 539-554. <http://eudml.org/doc/197590>.

@article{Miranville2010,
abstract = { We derive in this article some models of Cahn-Hilliard equations in nonisotropic media. These models, based on constitutive equations introduced by Gurtin in [19], take the work of internal microforces and also the deformations of the material into account. We then study the existence and uniqueness of solutions and obtain the existence of finite dimensional attractors. },
author = {Miranville, Alain},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Cahn-Hilliard equation; internal microforces; deformable continuum; nonisotropic material; global attractor; exponential attractor.; Cahn-Hillard equation; finite dimensional attractor; existence; uniqueness; initial value problem},
language = {eng},
month = {3},
number = {3},
pages = {539-554},
publisher = {EDP Sciences},
title = { Some models of Cahn-Hilliard equations in nonisotropic media },
url = {http://eudml.org/doc/197590},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Miranville, Alain
TI - Some models of Cahn-Hilliard equations in nonisotropic media
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 3
SP - 539
EP - 554
AB - We derive in this article some models of Cahn-Hilliard equations in nonisotropic media. These models, based on constitutive equations introduced by Gurtin in [19], take the work of internal microforces and also the deformations of the material into account. We then study the existence and uniqueness of solutions and obtain the existence of finite dimensional attractors.
LA - eng
KW - Cahn-Hilliard equation; internal microforces; deformable continuum; nonisotropic material; global attractor; exponential attractor.; Cahn-Hillard equation; finite dimensional attractor; existence; uniqueness; initial value problem
UR - http://eudml.org/doc/197590
ER -

References

top
  1. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of partial differential equations satisfying general boundary conditions I, II. Comm. Pure Appl. Math.12 (1959) 623-727 ; 17 (1964) 35-92.  Zbl0093.10401
  2. A. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain. J. Dyn. Differential Equations7 (1995) 567-590.  Zbl0846.35061
  3. A.V. Babin and M.I. Vishik, Attractors of evolution equations. North-Holland, Amsterdam (1991).  Zbl0804.58003
  4. H. Brezis, Analyse fonctionnelle, théorie et applications. Masson (1983).  Zbl0511.46001
  5. J.W. Cahn, On spinodal decomposition. Acta Metall.9 (1961) 795-801.  
  6. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys.2 (1958) 258-267.  
  7. M. Carrive, A. Miranville, A. Piétrus and J.M. Rakotoson, The Cahn-Hilliard equation for an isotropic deformable continuum. Appl. Math. Letters12 (1999) 23-28.  Zbl0939.35042
  8. M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic continua. Adv. Math. Sci. Appl. (to appear).  Zbl0987.35156
  9. V.V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension. J. Math. Pures Appl.73 (1994) 279-333.  Zbl0838.58021
  10. L. Cherfils and A. Miranville, Generalized Cahn-Hilliard equations with a logarithmic free energy (submitted).  Zbl1002.35062
  11. J.W. Cholewe and T. Dlotko, Global attractors of the Cahn-Hilliard system. Bull. Austral. Math. Soc.49 (1994) 277-302.  
  12. A. Debussche and L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. TMA24 (1995) 1491-1514.  Zbl0831.35088
  13. A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations. Masson (1994).  Zbl0842.58056
  14. M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in n with a strong nonlinearity. Disc. Cont. Dyn. Systems5 (1999) 399-424.  Zbl0959.35025
  15. C.M. Elliot and S. Luckhauss, A generalized equation for phase separation of a multi-component mixture with interfacial free energy. Preprint.  
  16. P. Fabrie and A. Miranville, Exponential attractors for nonautonomous first-order evolution equations. Disc. Cont. Dyn. Systems4 (1998) 225-240.  Zbl0980.34051
  17. C. Galusinski, Perturbations singulières de problèmes dissipatifs : étude dynamique via l'existence et la continuité d'attracteurs exponentiels. Thèse, Université Bordeaux-I (1996).  
  18. C. Galusinski, M. Hnid and A. Miranville, Exponential attractors for nonautonomous partially dissipative equations. Differential Integral Equations12 (1999) 1-22.  Zbl1012.35010
  19. M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D92 (1996) 178-192.  Zbl0885.35121
  20. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969).  
  21. D. Li and C. Zhong, Global attractor for the Cahn-Hilliard system with fast growing nonlinearity. J. Differential Equations (1998).  Zbl0912.35029
  22. M. Marion and R. Temam, Navier-Stokes equations, theory and approximation, in Handbook of numerical analysis, P.G. Ciarlet and J.L. Lions eds. (to appear).  Zbl0921.76040
  23. A. Miranville, Exponential attractors for nonautonomous evolution equations. Appl. Math. Letters11 (1998) 19-22.  Zbl06587011
  24. A. Miranville, Exponential attractors for a class of evolution equations by a decomposition method. C. R. Acad. Sci.328 (1999) 145-150.  Zbl1101.35334
  25. A. Miranville, Long time behavior of some models of Cahn-Hilliard equations in deformable continua. Nonlinear Anal. Series B (to appear).  Zbl0989.35066
  26. A. Miranville, Exponential attractors for a class of evolution equations by a decomposition method. II. The nonautonomous case. C. R. Acad. Sci.328 (1999) 907-912.  Zbl1141.35340
  27. A. Miranville, Equations de Cahn-Hilliard généralisées dans un milieu déformable. C. R. Acad. Sci.328 (1999) 1095-1100.  Zbl0930.35175
  28. A. Miranville, A model of Cahn-Hilliard equation based on a microforce balance. C. R. Acad. Sci.328 (1999) 1247-1252.  Zbl0932.35118
  29. A. Miranville, A. Piétrus and J.M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance. Asymptotic Anal.16 (1998) 315-345.  Zbl0936.35036
  30. B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations. Comm. Partial Differential Equations14 (1989) 245-297.  Zbl0691.35019
  31. R. Temam, Infinite dimensional dynamical systems in mechanics and physics. 2nd. ed., Springer-Verlag, New-York (1997).  Zbl0871.35001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.