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Global attractor for the Cahn-Hilliard system

Published online by Cambridge University Press:  17 April 2009

Jan W. Cholewa
Affiliation:
Institute of Mathematics Silesian University40-007 Katowice, Poland
Tomasz Dlotko
Affiliation:
Institute of Mathematics Silesian University40-007 Katowice, Poland
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The Cahn-Hilliard system, a natural extension of the single Cahn-Hilliard equation in the case of multicomponent alloys, will be shown to generate a dissipative semigroup on the phase space = [H2(Ω)]m. Following Hale's ideas and based on the existence and form of the Lyapunov functional, our main result will be the existence of a global attractor on a subset of . New difficulties specific to the system case make our problem interesting.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Cahn, J.W. and Hilliard, J.E., ‘Free energy of a nonuniform system, I. Interfacial free energy’, J. Chem. Phys. 28 (1958), 258267.CrossRefGoogle Scholar
[2]Dlotko, T., ‘Global attractor for the Cahn-Hilliard equation in H 2 and H 3’, J. Differential Equations (to appear).Google Scholar
[3]Elliot, C.M. and Luckhaus, S., ‘A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy’, IMA preprint 887, 1991.Google Scholar
[4]Eyre, D.J., ‘Systems of Cahn-Hilliard equations’, University of Minnesota, AHPCRC Preprint 92102, 1992.Google Scholar
[5]Hale, J.K., Asymptotic behavior of dissipative systems (American Mathematical Society, Providence, R.I., 1988).Google Scholar
[6]Henry, D., Geometric theory of semilinear parabolic equations (Springer-Verlag, Berlin, Heidelberg, New York, 1981).CrossRefGoogle Scholar
[7]Smoller, J., Shock waves and reaction-diffusion equations (Springer-Verlag, Berlin, Heidelberg, New York, 1988).Google Scholar
[8]Solonnikov, V.A., ‘On Lp estimates of solutions of elliptic and parabolic systems’, Trudy Mat. Inst. Steklov. 102 (1967), 137160.Google Scholar
[9]Szmydt, Z., Fourier transformation and linear differential equations (D. Reidel Publishing Company (PWN), Dordrecht, Holland, 1977).Google Scholar
[10]Temam, R., Infinite-dimensional dynamical systems in mechanics and physics (Springer-Verlag, Berlin, Heidelberg, New York, 1988).CrossRefGoogle Scholar
[11]Triebel, H., Interpolation theory, function spaces, differential operators (Veb Deutscher Verlag, Berlin 1978, also North-Holland, Amsterdam, 1978).Google Scholar
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