Global existence for a quasi-linear evolution equation with a non-convex energy

Feireisl, Eduard; Petzeltová, Hana

doi:10.1090/S0002-9947-01-02950-6

Global existence for a quasi-linear evolution equation with a non-convex energy
HTML articles powered by AMS MathViewer

by Eduard Feireisl and Hana Petzeltová

Trans. Amer. Math. Soc. 354 (2002), 1421-1434

DOI: https://doi.org/10.1090/S0002-9947-01-02950-6

Published electronically: December 5, 2001

PDF | Request permission

Abstract:

We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.

References

J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart, On the dynamics of fine structure, J. Nonlinear Sci. 1 (1991), no. 1, 17–70. MR 1102830, DOI 10.1007/BF01209147
Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
Ph. Clément, G. Gripenberg, and S.-O. Londen, Schauder estimates for equations with fractional derivatives, Trans. Amer. Math. Soc. 352 (2000), no. 5, 2239–2260. MR 1675170, DOI 10.1090/S0002-9947-00-02507-1
Ivar Ekeland and Roger Témam, Convex analysis and variational problems, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Translated from the French. MR 1727362, DOI 10.1137/1.9781611971088
Hans Engler, Global smooth solutions for a class of parabolic integrodifferential equations, Trans. Amer. Math. Soc. 348 (1996), no. 1, 267–290. MR 1321573, DOI 10.1090/S0002-9947-96-01472-9
Joachim Escher and Gieri Simonett (eds.), Topics in nonlinear analysis, Progress in Nonlinear Differential Equations and their Applications, vol. 35, Birkhäuser Verlag, Basel, 1999. The Herbert Amann anniversary volume. MR 1724787, DOI 10.1007/978-3-0348-8765-6
G. Friesecke and G. Dolzmann, Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy, SIAM J. Math. Anal. 28 (1997), no. 2, 363–380. MR 1434040, DOI 10.1137/S0036141095285958
G. Friesecke and J. B. McLeod, Dynamics as a mechanism preventing the formation of finer and finer microstructure, Arch. Rational Mech. Anal. 133 (1996), no. 3, 199–247. MR 1387930, DOI 10.1007/BF00380893
Herbert Gajewski, Konrad Gröger, and Klaus Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38, Akademie-Verlag, Berlin, 1974 (German). MR 0636412
G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
Robert L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal. 97 (1987), no. 4, 353–394. MR 865845, DOI 10.1007/BF00280411
H. Petzeltová and J. Prüss. Global stability of a fractional partial differential equation. J. Int. Equations Appl., 12(3):323–347, 2000.
Jan Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939, DOI 10.1007/978-3-0348-8570-6
Piotr Rybka, Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions, Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), no. 1-2, 101–138. MR 1169897, DOI 10.1017/S0308210500014177
Piotr Rybka, The viscous damping prevents propagation of singularities in the system of viscoelasticity, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 5, 1067–1074. MR 1475646, DOI 10.1017/S0308210500026895
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580