Attractors for gradient flows of nonconvex functionals and applications

Archive for Rational Mechanics and Analysis

Volumn 187, Issue 1, 2008, Pages 91-135

  Rossi, Riccarda ^a   Segatti, Antonio ^b   Stefanelli, Ulisse ^c  

^a UNIVERSITY OF BRESCIA   (Italy)

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

^c CNR   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 36549082550     PISSN: 00039527     EISSN: 14320673     Source Type: Journal    
DOI: 10.1007/s00205-007-0078-0     Document Type: Article

References (41)

1. Ambrosio L. (1995) Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19, 191-246
2. Ambrosio L., Gigli N., Savaré G., (2005) Gradient flows in metric spaces and in the Wasserstein spaces of probability measures Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel
3. Balder E.J. (1984) A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22(4): 570-598
4. Ball J.M.: A version of the fundamental theorem for Young measures. PDEs and continuum models of phase transitions (Nice 1988), Lecture Notes in Phys., vol. 344. pp. 207-215. Springer, Berlin, 1989
5. Ball J.M. (1997) Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7, 475-502
6. Ball J.M. (2004) Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10 1, 31-52
7. Blowey J.F., Elliott C.M. (1991) The Cahn-Hilliard gradient theory for the phase separations with nonsmooth free energy. I. Euro. J. Appl. Math. 2(3): 233-280
8. Brézis H.: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. Contribution to Nonlinear Functional Analysis, Proc. Sympos. Math. Res. Center, Univ. Wisconsin, Madison, 1971. Academic, New York, 1971
9. Brézis H. (1973) Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam
10. Brézis H., (1983) Analyse fonctionnelle - Théorie et applications. Masson, Paris
11. Brokate M., Sprekels J.: Hysteresis and phase transitions. Appl. Math. Sci., vol. 121. Springer, New York, 1996
12. Caginalp G. (1986) An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205-245
13. Caraballo T., Marin-Rubio P., Robinson J.C. (2003) A comparison between two theories for multi-valued semiflows and their asymptotic behaviour. Set-Valued Anal. 11(3): 297-322
14. Cardinali T., Colombo G., Papalini F., Tosques M. (1997) On a class of evolution equations without convexity. Nonlin. Anal. 282, 217-234
15. Chepyshoz V.V., Vishik M.I.: Attractors for equations of mathematical physics. American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, 2002
16. Crandall M.G., Pazy A. (1969) Semi-groups of nonlinear contractions and dissipative sets. J. Funct. Anal. 3, 376-418
17. De Giorgi E.: New problems on minimizing movements. Boundary Value Problems for PDE and Applications (Eds. Baiocchi C. and Louis Lions J.) Masson, Paris, 1993
18. De Giorgi E., Marino A., Tosques M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180-187 (1980)
19. Ould Elmounir A., Simondon F. (2000) Attracteurs compacts pour des problèmes d'evolution sans unicité. Ann. Fac. Sci. Toulouse Math. (6) 9(4): 631-654
20. Evans L.C., (1992) Gariepy R.: Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton
21. Hale J.K., Raugel G. (1988) Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. J. Differ. Equ. 73, 197-214
22. Kapustyan A.V., Melnik V.S., Valero J. (2003) Attractors of multivalued dynamical processes generated by phase-field equations. Int. J. Bifur. Chaos Appl. Sci. Eng. 13(7): 1969-1983
23. Kōmura Y.(1967) Nonlinear semi-groups in Hilbert space. J. Math. Soc. Jpn. 19, 493-507
24. Marino A., Saccon C., Tosques M. (1989) Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16(2): 281-330
25. Melnik V.S., Valero J. (1998) On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal. (4) 6, 83-111
26. Melnik V.S., Valero J. (2000) On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal. (4) 8, 375-403
27. Plotnikov P.I., Starovoitov V.N. (1993) The Stefan problem with surface tension as the limit of a phase field model. Differential Equations 29, 395-404
28. Rocca E., Schimperna G. (2004) Universal attractor for some singular phase transition systems. Phys. D 192(3-4): 279-307
29. Rossi R., Savaré G. (2004) Existence and approximation results for gradient flows. Rend. Mat. Acc. Lincei 15, 183-196
30. Rossi R., Savaré G. (2006) Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM Control Optim. Calc. Var. 12(3): 564-614
31. Rossi R., Segatti A., Stefanelli U.: Attractors for gradient flows of non convex functionals and applications. Preprint IMATI-CNR n. 6-PV, 1-47, 2006
32. Schätzle R. (2000) The quasistationary phase field equations with Neumann boundary conditions J. Differential Equations 162(2): 473-503
33. Sell G.R. (1973) Differential equations without uniqueness and classical topological dynamics. J. Differ. Equ. 14, 42-56
34. Sell G.R. (1996) Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Differ. Equ. 8(1): 1-33
35. Segatti A. (2006) Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete Contin. Dyn. Syst. 14(4): 801-820
36. Segatti A. (2007) On the hyperbolic relaxation of the Cahn-Hilliard equation in 3-D: approximation and long time behaviour. Math. Models. Methods Appl. Sci. 17(3): 411-437
37. Shirakawa K., Ito A., Yamazaki N., Kenmochi N.: Asymptotic stability for evolution equations governed by subdifferentials. Recent developments in domain decomposition methods and flow problems (Kyoto, 1996; Anacapri, 1996). GAKUTO Internat. Ser. Math. Sci. Appl., vol 11. Gakkōtosho Tokyo, 1998
38. p (0,T;B). Ann. Mat. Pura Appl. (4) 146, 65-96
39. Temam R., (1988) Infinite dimensional mechanical systems in mechanics and physics. Applied Mathematical Sciences, vol 68. Springer, New York
40. Valero J. (2001) Attractors for parabolic equations without uniqueness. J. Dyn. Differ. Equ. 13, 711-744
41. Visintin A.: Models of phase transitions, Progress in Nonlinear Differential Equations and Their Applications, vol. 28. Birkhäuser, Boston, 1996