Skip to main content
×
Home
    • Aa
    • Aa

Long-time behaviour for a model of phase-field type

  • Ph. Laurençot (a1)
Abstract

In this paper, we study a model of phase-field type for the kinetics of phase transitions which was considered by Halperin, Hohenberg and Ma and which includes the phase-field equations. We study the well-posedness of the corresponding initial boundary value problem in an open bounded subset in space dimension lower than or equal to 3 and prove that, under suitable conditions, the long-time behaviour of the solutions to this problem is described by a maximal attractor.

Copyright
References
Hide All
1Adams R. A.. Sobolev Spaces (New York: Academic Press, 1975).
2Amann H.. Dynamic theory of quasilinear parabolic equations, II. Reaction-diffusion systems. Differential Integral Equations 3 (1990), 1375.
3Amann H.. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function Spaces, Differential Operators and Nonlinear Analysis, eds Triebel H. and Schmeisser H. J. Teubner-Texte Math. 133, 9126 (Stuttgart: Teubner, 1993).
4Bates P. W. and Zheng S.. Inertial manifolds and inertial sets for phase-field equations. J. Dynamics Differential Equations 4 (1992), 375–97.
5Brochet D., Chen X. and Hilhorst D.. Finite dimensional exponential attractor for the phase field model. Appl. Anal. 49 (1993), 197212.
6Brochet D. and Hilhorst D.. Universal attractor and inertial sets for the phase field model. Appl. Math. Lett. 4(1991), 5962.
7Caginalp G.. An analysis of a phase-field model of a free boundary. Arch. Rational Mech. Anal. 92 (1986), 205–45.
8Eden A., Foias C., Nicolaenko B. and Temam R.. Ensembles inertiels pour des equations devolution dissipatives. C. R. Acad. Sci. Paris I 310 (1990), 559–62.
9Eden A., Foias C., Nicolaenko B. and Temam R.. Inertial sets for dissipative evolution equations, Part I: construction and application (IMA Preprint Series 812, 1991).
10Elliott C. M. and Zheng S.. Global existence and stability of solutions to the phase field equations. In Free Boundary Value Problems, eds Hoffmann K. H. and Sprekels J., International Series of Numerical Mathematics 95, 4658 (Basel: Birkhauser, 1990).
11Halperin B. I., Hohenberg P. C. and Ma S. K.. Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation. Phys. Rev. B 10 (1974), 139–53.
12Haraux A.. Systemes Dynamiques Dissipatifs et Applications (Paris: Masson, 1990).
13Penrose O. and Fife P. C.. Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Phys. D 43 (1990), 4462.
14Temam R.. Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68 (New York: Springer, 1988).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 35 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.