Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy

SIAM Journal on Mathematical Analysis

Volumn 28, Issue 2, 1997, Pages 363-380

  Friesecke, G ^a   Dolzmann, G ^b  

^a UNIVERSITY OF FREIBURG   (Germany)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

Evolution equations; Implicit time discretization; Nonconvex functionals; Solid solid phase transitions; Viscoelasticity

Indexed keywords

EID: 0031488286     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141095285958     Document Type: Article

References (39)

1. x)x, J. Differential Equations, 35 (1980), pp. 200-231.
2. G. ANDREWS AND J. M. BALL, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations, 44 (1982), pp. 306-341.
3. H. W. ALT, K.-H. HOFFMANN, M. NIEZGÓDKA, AND J. SPREKELS, A numerical study of structural phase transitions in shape memory alloys, preprint, Institut für Mathematik, Universität Augsburg, Augsburg, Germany, 1985.
4. H. BELLOUT, F. BLOOM, AND J. NEČAS, Existence of global weak solutions to the dynamical problem for a three-dimensional elastic body with singular memory, SIAM J. Math. Anal., 24 (1993), pp. 36-45.
5. H. BELLOUT, F. BLOOM, AND J. NEČAS, Solutions for incompressible non-Newtonian fluids, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), pp. 795-800.
6. D. BRANDON, I. FONSECA, AND P. J. SWART, The creation and propagation of oscillations in a dynamical model of displacive phase, transformations, to appear.
7. 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), pp. 17-70.
8. J. M. BALL AND R. D. JAMES, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), pp. 13-52.
9. J. M. BALL AND R. D. JAMES, Proposed experimental tests of a theory of fine microstructure, and the two-well problem, Philos. Trans. Roy. Soc. London, Ser. A, 338 (1992), pp. 389-450.
10. H. BELLOUT AND J. NEČAS, Existence of global weak solutions for a class of quasilinear hyperbolic integro-differential equations describing visco-elastic materials, Math. Ann., 299 (1994), pp. 275-291.
11. J. CLEMENT, Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26 (1974), pp. 745-752.
12. Z. CHEN AND K.-H. HOFFMANN, On a one-dimensional nonlinear thermoviscoelastic model for structural phase transitions in shape memory alloys, J. Differential Equations, 112 (1994), pp. 325-350.
13. M. CHIPOT AND D. KINDERLEHRER, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., 103 (1988), pp. 237-277.
14. B. DACOROGNA, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989.
15. C. M. DAFERMOS, The mixed initial-boundary value problem for the equations of non-linear one dimensional viscoelasticity, J. Differential Equations, 6 (1969), pp. 71-86.
16. S. DEMOULINI, Young measure solutions for nonlinear systems of evolution, preprint, 1994.
17. H. ENGLER, Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Math. Z., 202 (1989), pp. 251-259.
18. H. ENGLER, Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials, Arch. Rational Mech. Anal., 113 (1991), pp. 1-38.
19. G. FRIESECKE, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), pp. 437-471.
20. G. FRIESECKE AND J. B. MCLEOD, Dynamics as a mechanism preventing the formation of finer and finer microstructure, Arch. Rational Mech. Anal., 133 (1996), pp. 199-247.
21. A. FRIEDMAN AND J. NEČAS, Systems of nonlinear wave equations with nonlinear viscosity, Pacific J. Math., 132 (1988), pp. 29-55.
22. H. GAJEWSKI, K. GRÖGER, AND K. ZACHARIAS, Nichtlineare Operatoryleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
23. tt, J. Math. Mech., 17 (1968), pp. 707-728.
24. K.-H. HOFFMANN AND A. ZOCHOWSKI, Analysis of the thermoelastic model of a plate with non-linear shape memory reinforcements, Math. Methods Appl. Sci., 15 (1992), pp. 631-645.
25. W. D. KALIES AND P. J. HOLMES, On a dynamical model for phase transformation in nonlinear elasticity, preprint, 1993.
26. P. KLOUČEK AND M. LUSKIN, Computational modeling of the martensitic transformation with surface energy, Math. Comput. Modelling, to appear.
27. P. KLOUČEK AND M. LUSKIN, The computation of the dynamics of the martensitic transformation, Contin. Mech. Thermodyn., to appear.
28. D. KINDERLEHRER AND P. PEDREGAL, Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal., 22 (1992), pp. 1-19.
29. J.-L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.
30. J. MILOTA, J. NEČAS, AND V. ŠVERÁK, On weak solutions to a viscoelastic model, Comment. Math. Univ. Carolin., 31 (1990), pp. 557-565.
31. J. NEČAS AND M. RUŽIČKA, Global solutions to the incompressible viscous-multipolar material, 1989, preprint.
32. M. NIEZGODKA AND J. SPREKELS, Existence of solutions of a mathematical model of structural phase transitions in shape memory alloys, Math. Methods Appl. Sci., 10 (1988), pp. 197-223.
33. R. L. PEGO, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal., 97 (1987), pp. 353-394.
34. H. PECHER, On global regular solutions of third order partial differential equations, J. Math. Anal. Appl., 73 (1980), pp. 278-299.
35. P. RYBKA, Dynamical modeling of phase transitions by means of viscoelasticity in many dimensions, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101-138.
36. P. SWART, The dynamical creation of microstructure in material phase transitions, Dissertation, Cornell University, Ithaca, NY, 1991.
37. P. J. SWART AND P. J. HOLMES, Energy minimization and the formation of micro structure in dynamic anti-plane shear, Arch. Rational Mech. Anal., 121 (1992), pp. 37-85.
38. J. SPREKELS AND Z. SONGMU, Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys, Phys. D., 39 (1989), pp. 59-76.
39. xtx = f, Osaka J. Math., 17 (1980), pp. 303-323.