A descent method for the free energy of multicomponent systems

Discrete and Continuous Dynamical Systems

Volumn 15, Issue 2, 2006, Pages 505-528

  Gajewski, Herbert ^a   Griepentrog, Jens A ^a  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Asymptotic behaviour; Cahn Hilliard equation; Image segmentation; Lyapunov function; Nonconvex functional; Phase separation

Indexed keywords

EID: 33745316599     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2006.15.505     Document Type: Article

References (17)

1. Bates, P.W., Chmaj, A.: An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions. J. Statist. Phys. 95, 1119-1139 (1999).
2. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258-267 (1958).
3. Chen, C.-K., Fife, P.C.: Nonlocal models of phase transitions in solids. Adv. Math. Sci. Appl. 10, 821-849 (2000).
4. Chill, R.: On the Lojasiewicz-Simon gradient inequality. J. Funct. Anal. 201, 572-601 (2003).
5. Dieudonné, J.; Grundzüge der modernen Analysis, Band 1. Dritte Auflage. Braunschweig, Wiesbaden: Vieweg & Sohn, 1985.
6. Feireisl, E., Issard-Roch, F., Petzeltova, H.: Long-time behaviour and convergence towards equilibria for a conserved phase field model. Discrete Contin. Dynam. Systems 10, 239-252 (2004).
7. Feireisl, E., Issard-Roch, F., Petzeltova, H.: A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems. J. Differential Equations 199, 1-21 (2004).
8. Giacomin, G., Lebowitz, J.L., Marra, R.: Macroscopic evolution of particle systems with short- and long-range interactions. Nonlinearity 13, 2143-2162 (2000).
9. Gajewski, H., Zacharias, K.: On a nonlocal phase separation model. J. Math. Anal. Appl. 286, 11-31 (2003).
10. Gajewski, H., Gärtner, K.: On a nonlocal model of image segmentation. Z. Angew. Math. Phys. 56, 572-591 (2005).
11. Giusti, E.: Metodi diretti nel calcolo delle variazioni. Bologna: Unione Matematica Italiana, 1994.
12. Griepentrog, J.A.: Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals. Math. Nachr. 843, 19-42 (2002).
13. Griepentrog, J.A.: On the unique solvability of a nonlocal phase separation problem for multicomponent systems. Banach Center Publ. 66, 153-164 (2004).
14. Lojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles 117. Paris: Editions du C.N.R.S., 87-89 (1963).
15. Miranville, A., Rougirel, A.: Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equation. Preprint, Université de Poitiers, 1-24 (2004).
16. Morel, J.-M., Solimini, S.: Variational Methods in Image Segmentation. Boston, Basel, Berlin: Birkhäuser, 1994.
17. Wainberg, M.M., Trenogin, W.A.: Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen. Berlin: Akademie-Verlag, 1973.