Global solvability of a dissipative Frémond model for shape memory alloys. II. Existence

Bonetti, Elena

doi:10.1090/qam/2032572

Author: Elena Bonetti
Journal: Quart. Appl. Math. 62 (2004), 53-76
MSC: Primary 74N99; Secondary 35K85, 35Q72, 74H15, 74H20
DOI: https://doi.org/10.1090/qam/2032572
MathSciNet review: MR2032572
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Abstract: The paper investigates an initial and boundary values problem which is derived from a dissipative Frémond model for shape memory alloys. Existence of a global solution for the abstract version of the evolution problem is proved by use of a semi-implicit time discretization scheme combined with an a priori estimates-passage to the limit procedure.

Sergiu Aizicovici, Pierluigi Colli, and Maurizio Grasselli, Doubly nonlinear evolution equations with memory, Funkcial. Ekvac. 44 (2001), no. 1, 19–51. MR 1847835
Claudio Baiocchi, Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl. (4) 76 (1967), 233–304 (Italian). MR 223697, DOI https://doi.org/10.1007/BF02412236
Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843

H. Brézis, Opérateurs maximaux monotones et semi-groups de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973

Elena Bonetti, Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness, Quart. Appl. Math. 61 (2003), no. 4, 759–781. MR 2019622, DOI https://doi.org/10.1090/qam/2019622
Elena Bonetti, Global solution to a Frémond model for shape memory alloys with thermal memory, Nonlinear Anal. 46 (2001), no. 4, Ser. A: Theory Methods, 535–565. MR 1856593, DOI https://doi.org/10.1016/S0362-546X%2800%2900131-0
N. Chemetov, Uniqueness results for the full Frémond model of shape memory alloys, Z. Anal. Anwendungen 17 (1998), no. 4, 877–892. MR 1669913, DOI https://doi.org/10.4171/ZAA/856
Pierluigi Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Anal. 24 (1995), no. 11, 1565–1579. MR 1328584, DOI https://doi.org/10.1016/0362-546X%2894%2900097-2
Pierluigi Colli, Michel Frémond, and Augusto Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math. 48 (1990), no. 1, 31–47. MR 1040232, DOI https://doi.org/10.1090/qam/1040232

M. Frémond, Matériaux a mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers. Sci. Terre. 304, 239-244 (1987)

M. Frémond, Shape memory alloys. A thermomechanical model, in: Free Boundary Problems: Theory and applications, vol. I–II (K. H. Hoffmann and J. Sprekels, eds.), Pitman Res. Notes Math. Ser. 185, Longman, London, 1990

M. Frémond, Non smooth thermo-mechanics, Springer-Verlag, 2001

M. Frémond and S. Miyazaki, Shape memory alloys, in: CISM Courses and Lectures No. 351, Springer-Verlag, New York, 1996

Joseph W. Jerome, Approximation of nonlinear evolution systems, Mathematics in Science and Engineering, vol. 164, Academic Press, Inc., Orlando, FL, 1983. MR 690582
Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
Jürgen Sprekels, Shape memory alloys: mathematical models for a class of first order solid-solid phase transitions in metals, Control Cybernet. 19 (1990), no. 3-4, 287–308 (1991). Optimal design and control of structures (Jabłonna, 1990). MR 1118688