On certain convex compactifications for relaxation in evolution problems

Tomáš Roubíček

doi:10.3934/dcdss.2011.4.467

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April 2011, 4(2): 467-482. doi: 10.3934/dcdss.2011.4.467

On certain convex compactifications for relaxation in evolution problems

Tomáš Roubíček ^1,

1.	Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8

Received March 2009 Revised July 2009 Published November 2010

Abstract
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A general-topological construction of limits of inverse systems is applied to convex compactifications and furthermore to special convex compactifications of Lebesgue-space-valued functions parameterized by time. Application to relaxation of quasistatic evolution in phase-change-type problems is outlined.

Keywords: generalized Young measures, Limits of inverse systems, Helly principle..

Mathematics Subject Classification: Primary: 54D35; Secondary: 26A45, 49J4.

Citation: Tomáš Roubíček. On certain convex compactifications for relaxation in evolution problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 467-482. doi: 10.3934/dcdss.2011.4.467

References:

[1]	P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension, Math. Anal., 30 (1929), 101-187. Google Scholar
[2]	S. Aubri, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials, Comp. Meth. in Appl. Mech. Engr., 192 (2003), 2823-2843. Google Scholar
[3]	V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces," D. Reidel Publ., Dordrecht, 1986. Google Scholar
[4]	S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143-5175. doi: 10.1016/j.cma.2003.12.065. Google Scholar
[5]	S. A. Belov and V. V. Chistyakov, A selection principle or mappings of bounded variation, J. Math. Anal. Appl., 249 (2000), 351-366. doi: 10.1006/jmaa.2000.6844. Google Scholar
[6]	F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measure approach,, SISSA, (). Google Scholar
[7]	C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, Proc. Royal Soc. London, Ser. A, 458 (2002), 299-317. Google Scholar
[8]	V. V. Chistyakov, Mappings of bounded variations, J. Dyn. Cont. Syst., 3 (1997), 261-289. doi: 10.1007/BF02465896. Google Scholar
[9]	V. V. Chistyakov and O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$, J. Dyn. Cont. Syst., 4 (1998), 217-247. doi: 10.1023/A:1022889902536. Google Scholar
[10]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures, Netw. Heterog. Media, 2 (2007), 1-36. Google Scholar
[11]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Rational Mech. Anal., 189 (2007), 469-544. doi: 10.1007/s00205-008-0117-5. Google Scholar
[12]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening, Netw. Heterog. Media, 3 (2008), 567-614. Google Scholar
[13]	R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: 10.1007/BF01214424. Google Scholar
[14]	S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology," Princeton, 1952. Google Scholar
[15]	R. Engelking, "General Topology," PWN, Warszawa, 1977. Google Scholar
[16]	A. Fiaschi, A Young measure approach to a quasistatic evolution for a class of material models with nonconvex elastic energies, ESAIM: Control, Optimisation Calc. Var., 15 (2009), 245-278. doi: 10.1051/cocv:2008030. Google Scholar
[17]	A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy, Ann. Inst. H. Poincaré, Anal. Nonlin., 26 (2009), 1055-1080. Google Scholar
[18]	A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, (). Google Scholar
[19]	S. Govindjee, A. Mielke and G. J. Hall, Free-energy of mixing for $n$-variant martensitic phase transformations using quasi-convex analysis, J. Mech. Physics Solids, 50 (2002), 1897-1922. doi: 10.1016/S0022-5096(02)00009-1. Google Scholar
[20]	K. Hackl and D. M. Kochmann, Relaxed potentials and evolution equations for inelastic microstructures, in "IUTAM Symp. Theoretical, Comput. Model. Aspects of Inelastic Media" (B. D. Reddy, ed.), Springer, (2008), 27-39. doi: 10.1007/978-1-4020-9090-5_3. Google Scholar
[21]	B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés. J. Mécanique, 14 (1975), 39-63. Google Scholar
[22]	E. Helly, Über lineare funktionaloperationen, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265-297. Google Scholar
[23]	M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1. Google Scholar
[24]	M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev., 48 (2006), 439-483. doi: 10.1137/S0036144504446187. Google Scholar
[25]	M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces, (preprint no. 5/07, BICS, Univ. Bath, 2007), ESAIM Conv. Optim. Calc. Var. doi: 10.1051/cocv:2008060. Google Scholar
[26]	M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity. (preprint no. 20/08, BICS, Uni. Bath, 2008), submitted. Google Scholar
[27]	M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures, (preprint no. 19/08, BICS, Uni. Bath, 2008), submitted. Google Scholar
[28]	S. Lefschetz, On compact spaces, Math. Anal., 32 (1931), 521-538. doi: 10.2307/1968249. Google Scholar
[29]	A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. PDEs, 22 (2005), 73-99. Google Scholar
[30]	A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5095-5127. doi: 10.1016/j.cma.2004.07.003. Google Scholar
[31]	A. Mielke, Evolution of rate-independent systems, in "Handbook of Differential Equations: Evolutionary Diff. Eqs." (C. Dafermos and E. Feireisl, eds.), North-Holland, Elsevier, Amsterdam, (2005), 461-559. Google Scholar
[32]	A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case, in "Multifield Problems in Solid and Fluid Mechanics" (R. Helmig, A. Mielke and B. I. Wohlmuth, eds.), L.N. Appl. Comp. Mech., 28, Springer, Berlin, (2006), 351-379. Google Scholar
[33]	A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860. Google Scholar
[34]	A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189, (accepted July 2001). Google Scholar
[35]	A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Archive Rat. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194. Google Scholar
[36]	T. Roubíček, Convex compactifications and special extensions of optimization problems, Nonlinear Anal., Th. Meth. Appl., 16 (1991), 1117-1126. doi: 10.1016/0362-546X(91)90199-B. Google Scholar
[37]	T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus," W. de Gruyter, Berlin, 1997. Google Scholar
[38]	T. Roubíček, Convex locally compact extensions of Lebesgue spaces and their applications, in: "Calculus of Variations and Optimal Control" (A. Ioffe, S. Reich and I. Shafrir, eds.), Chapman & Hall/CRC, Boca Raton, FL, (1999), 237-250. Google Scholar
[39]	T. Roubíček and K.-H. Hoffmann, About the concept of measure-valued solutions to distributed parameter systems, Math. Methods Appl. Sci., 18 (1995), 671-685. doi: 10.1002/mma.1670180902. Google Scholar
[40]	T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics, Z. angew. Math. Physik, 55 (2004), 159-182. Google Scholar
[41]	A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Annalen, 102 (1930), 544-561. doi: 10.1007/BF01782364. Google Scholar

show all references

References:

[1]	P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension, Math. Anal., 30 (1929), 101-187. Google Scholar
[2]	S. Aubri, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials, Comp. Meth. in Appl. Mech. Engr., 192 (2003), 2823-2843. Google Scholar
[3]	V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces," D. Reidel Publ., Dordrecht, 1986. Google Scholar
[4]	S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143-5175. doi: 10.1016/j.cma.2003.12.065. Google Scholar
[5]	S. A. Belov and V. V. Chistyakov, A selection principle or mappings of bounded variation, J. Math. Anal. Appl., 249 (2000), 351-366. doi: 10.1006/jmaa.2000.6844. Google Scholar
[6]	F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measure approach,, SISSA, (). Google Scholar
[7]	C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity, Proc. Royal Soc. London, Ser. A, 458 (2002), 299-317. Google Scholar
[8]	V. V. Chistyakov, Mappings of bounded variations, J. Dyn. Cont. Syst., 3 (1997), 261-289. doi: 10.1007/BF02465896. Google Scholar
[9]	V. V. Chistyakov and O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$, J. Dyn. Cont. Syst., 4 (1998), 217-247. doi: 10.1023/A:1022889902536. Google Scholar
[10]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures, Netw. Heterog. Media, 2 (2007), 1-36. Google Scholar
[11]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Rational Mech. Anal., 189 (2007), 469-544. doi: 10.1007/s00205-008-0117-5. Google Scholar
[12]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening, Netw. Heterog. Media, 3 (2008), 567-614. Google Scholar
[13]	R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: 10.1007/BF01214424. Google Scholar
[14]	S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology," Princeton, 1952. Google Scholar
[15]	R. Engelking, "General Topology," PWN, Warszawa, 1977. Google Scholar
[16]	A. Fiaschi, A Young measure approach to a quasistatic evolution for a class of material models with nonconvex elastic energies, ESAIM: Control, Optimisation Calc. Var., 15 (2009), 245-278. doi: 10.1051/cocv:2008030. Google Scholar
[17]	A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy, Ann. Inst. H. Poincaré, Anal. Nonlin., 26 (2009), 1055-1080. Google Scholar
[18]	A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, (). Google Scholar
[19]	S. Govindjee, A. Mielke and G. J. Hall, Free-energy of mixing for $n$-variant martensitic phase transformations using quasi-convex analysis, J. Mech. Physics Solids, 50 (2002), 1897-1922. doi: 10.1016/S0022-5096(02)00009-1. Google Scholar
[20]	K. Hackl and D. M. Kochmann, Relaxed potentials and evolution equations for inelastic microstructures, in "IUTAM Symp. Theoretical, Comput. Model. Aspects of Inelastic Media" (B. D. Reddy, ed.), Springer, (2008), 27-39. doi: 10.1007/978-1-4020-9090-5_3. Google Scholar
[21]	B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés. J. Mécanique, 14 (1975), 39-63. Google Scholar
[22]	E. Helly, Über lineare funktionaloperationen, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265-297. Google Scholar
[23]	M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1. Google Scholar
[24]	M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev., 48 (2006), 439-483. doi: 10.1137/S0036144504446187. Google Scholar
[25]	M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces, (preprint no. 5/07, BICS, Univ. Bath, 2007), ESAIM Conv. Optim. Calc. Var. doi: 10.1051/cocv:2008060. Google Scholar
[26]	M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity. (preprint no. 20/08, BICS, Uni. Bath, 2008), submitted. Google Scholar
[27]	M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures, (preprint no. 19/08, BICS, Uni. Bath, 2008), submitted. Google Scholar
[28]	S. Lefschetz, On compact spaces, Math. Anal., 32 (1931), 521-538. doi: 10.2307/1968249. Google Scholar
[29]	A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. PDEs, 22 (2005), 73-99. Google Scholar
[30]	A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5095-5127. doi: 10.1016/j.cma.2004.07.003. Google Scholar
[31]	A. Mielke, Evolution of rate-independent systems, in "Handbook of Differential Equations: Evolutionary Diff. Eqs." (C. Dafermos and E. Feireisl, eds.), North-Holland, Elsevier, Amsterdam, (2005), 461-559. Google Scholar
[32]	A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case, in "Multifield Problems in Solid and Fluid Mechanics" (R. Helmig, A. Mielke and B. I. Wohlmuth, eds.), L.N. Appl. Comp. Mech., 28, Springer, Berlin, (2006), 351-379. Google Scholar
[33]	A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860. Google Scholar
[34]	A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189, (accepted July 2001). Google Scholar
[35]	A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Archive Rat. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194. Google Scholar
[36]	T. Roubíček, Convex compactifications and special extensions of optimization problems, Nonlinear Anal., Th. Meth. Appl., 16 (1991), 1117-1126. doi: 10.1016/0362-546X(91)90199-B. Google Scholar
[37]	T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus," W. de Gruyter, Berlin, 1997. Google Scholar
[38]	T. Roubíček, Convex locally compact extensions of Lebesgue spaces and their applications, in: "Calculus of Variations and Optimal Control" (A. Ioffe, S. Reich and I. Shafrir, eds.), Chapman & Hall/CRC, Boca Raton, FL, (1999), 237-250. Google Scholar
[39]	T. Roubíček and K.-H. Hoffmann, About the concept of measure-valued solutions to distributed parameter systems, Math. Methods Appl. Sci., 18 (1995), 671-685. doi: 10.1002/mma.1670180902. Google Scholar
[40]	T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics, Z. angew. Math. Physik, 55 (2004), 159-182. Google Scholar
[41]	A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Annalen, 102 (1930), 544-561. doi: 10.1007/BF01782364. Google Scholar

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