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

On certain convex compactifications for relaxation in evolution problems

1. 

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

Received  March 2009 Revised  July 2009 Published  November 2010

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.
Citation: Tomáš Roubíček. On certain convex compactifications for relaxation in evolution problems. Discrete and 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.

[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.

[3]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces," D. Reidel Publ., Dordrecht, 1986.

[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.

[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.

[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, (). 

[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.

[8]

V. V. Chistyakov, Mappings of bounded variations, J. Dyn. Cont. Syst., 3 (1997), 261-289. doi: 10.1007/BF02465896.

[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.

[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.

[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.

[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.

[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.

[14]

S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology," Princeton, 1952.

[15]

R. Engelking, "General Topology," PWN, Warszawa, 1977.

[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.

[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.

[18]

A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, (). 

[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.

[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.

[21]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés. J. Mécanique, 14 (1975), 39-63.

[22]

E. Helly, Über lineare funktionaloperationen, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265-297.

[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.

[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.

[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.

[26]

M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity. (preprint no. 20/08, BICS, Uni. Bath, 2008), submitted.

[27]

M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures, (preprint no. 19/08, BICS, Uni. Bath, 2008), submitted.

[28]

S. Lefschetz, On compact spaces, Math. Anal., 32 (1931), 521-538. doi: 10.2307/1968249.

[29]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. PDEs, 22 (2005), 73-99.

[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.

[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.

[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.

[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.

[34]

A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189, (accepted July 2001).

[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.

[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.

[37]

T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus," W. de Gruyter, Berlin, 1997.

[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.

[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.

[40]

T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics, Z. angew. Math. Physik, 55 (2004), 159-182.

[41]

A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Annalen, 102 (1930), 544-561. doi: 10.1007/BF01782364.

show all references

References:
[1]

P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension, Math. Anal., 30 (1929), 101-187.

[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.

[3]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces," D. Reidel Publ., Dordrecht, 1986.

[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.

[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.

[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, (). 

[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.

[8]

V. V. Chistyakov, Mappings of bounded variations, J. Dyn. Cont. Syst., 3 (1997), 261-289. doi: 10.1007/BF02465896.

[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.

[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.

[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.

[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.

[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.

[14]

S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology," Princeton, 1952.

[15]

R. Engelking, "General Topology," PWN, Warszawa, 1977.

[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.

[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.

[18]

A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, (). 

[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.

[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.

[21]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés. J. Mécanique, 14 (1975), 39-63.

[22]

E. Helly, Über lineare funktionaloperationen, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265-297.

[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.

[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.

[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.

[26]

M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity. (preprint no. 20/08, BICS, Uni. Bath, 2008), submitted.

[27]

M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures, (preprint no. 19/08, BICS, Uni. Bath, 2008), submitted.

[28]

S. Lefschetz, On compact spaces, Math. Anal., 32 (1931), 521-538. doi: 10.2307/1968249.

[29]

A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. PDEs, 22 (2005), 73-99.

[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.

[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.

[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.

[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.

[34]

A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189, (accepted July 2001).

[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.

[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.

[37]

T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus," W. de Gruyter, Berlin, 1997.

[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.

[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.

[40]

T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics, Z. angew. Math. Physik, 55 (2004), 159-182.

[41]

A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Annalen, 102 (1930), 544-561. doi: 10.1007/BF01782364.

[1]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1

[2]

Kazuhiro Kawamura. Mean dimension of shifts of finite type and of generalized inverse limits. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4767-4775. doi: 10.3934/dcds.2020200

[3]

Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465

[4]

Steffen Arnrich. Modelling phase transitions via Young measures. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29

[5]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17 (2) : 227-254. doi: 10.3934/nhm.2022004

[6]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432

[7]

Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967

[8]

Zeya Mi. SRB measures for some diffeomorphisms with dominated splittings as zero noise limits. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6441-6465. doi: 10.3934/dcds.2019279

[9]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246

[10]

Chris Good, Robin Knight, Brian Raines. Countable inverse limits of postcritical $w$-limit sets of unimodal maps. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1059-1078. doi: 10.3934/dcds.2010.27.1059

[11]

Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51

[12]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109

[13]

Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233

[14]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control and Related Fields, 2021, 11 (4) : 829-855. doi: 10.3934/mcrf.2020048

[15]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems and Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

[16]

Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041

[17]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043

[18]

Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226

[19]

George Ballinger, Xinzhi Liu. Boundedness criteria in terms of two measures for impulsive systems. Conference Publications, 1998, 1998 (Special) : 79-88. doi: 10.3934/proc.1998.1998.79

[20]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]