BV solutions and viscosity approximations of rate-independent systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 1, p. 36-80

In the nonconvex case, solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate-independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential that is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of “BV solutions” involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play crucial roles in the description of the associated jump trajectories. We shall prove general convergence results for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.

DOI : https://doi.org/10.1051/cocv/2010054
Classification:  49Q20,  58E99
Keywords: doubly nonlinear, differential inclusions, generalized gradient flows, viscous regularization, vanishing-viscosity limit, vanishing-viscosity contact potential, parameterized solutions
@article{COCV_2012__18_1_36_0,
     author = {Mielke, Alexander and Rossi, Riccarda and Savar\'e, Giuseppe},
     title = {BV solutions and viscosity approximations of rate-independent systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {1},
     year = {2012},
     pages = {36-80},
     doi = {10.1051/cocv/2010054},
     zbl = {1250.49041},
     mrnumber = {2887927},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2012__18_1_36_0}
}
Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe. BV solutions and viscosity approximations of rate-independent systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 1, pp. 36-80. doi : 10.1051/cocv/2010054. http://www.numdam.org/item/COCV_2012__18_1_36_0/

[1] L. Ambrosio, Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR 1387558 | Zbl 0957.49029

[2] L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108 (1990) 691-702. | MR 969514 | Zbl 0685.49027

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000). | MR 1857292 | Zbl 0957.49001

[4] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, 2nd edn., Birkhäuser Verlag, Basel (2008). | MR 2401600 | Zbl 1145.35001

[5] F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Meth. Appl. Sci. 18 (2008) 125-164. | MR 2378086 | Zbl 1151.74319

[6] G. Bouchitté, A. Mielke and T. Roubíček, A complete-damage problem at small strains. Z. Angew. Math. Phys. 60 (2009) 205-236. | MR 2486153 | Zbl 1238.74005

[7] M. Buliga, G. De Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws. J. Convex Anal. 15 (2008) 87-104. | MR 2389005 | Zbl 1133.49018

[8] P. Colli, On some doubly nonlinear evolution equations in Banach spaces. Japan J. Indust. Appl. Math. 9 (1992) 181-203. | MR 1170721 | Zbl 0757.34051

[9] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15 (1990) 737-756. | MR 1070845 | Zbl 0707.34053

[10] G. Dal Maso and R. Toader, A model for quasi-static growth of brittle fractures : existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101-135. | MR 1897378 | Zbl 1042.74002

[11] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Meth. Appl. Sci. 12 (2002) 1773-1799. | MR 1946723 | Zbl 1205.74149

[12] G. Dal Maso and C. Zanini, Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. R. Soc. Edinb., Sect. A, Math. 137 (2007) 253-279. | MR 2360770 | Zbl 1116.74004

[13] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176 (2005) 165-225. | MR 2186036 | Zbl 1064.74150

[14] G. Dal Maso, A. Desimone and M.G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237-291. | MR 2210910 | Zbl 1093.74007

[15] 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. | MR 2425074 | Zbl 1156.74308

[16] G. Dal Maso, A. Desimone, M.G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189 (2008) 469-544. | MR 2424994 | Zbl 1219.35305

[17] G. Dal Maso, A. Desimone and F. Solombrino, Quasistatic evolution for cam-clay plasiticity : a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differential Equations (to appear). | MR 2745199 | Zbl pre05837762

[18] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Analysis 13 (2006) 151-167. | MR 2211809 | Zbl 1109.74040

[19] A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy. Ann. Inst. Henri Poincaré, Anal. Non Linéaire (to appear). | Numdam | Zbl 1167.74005

[20] G. Francfort and A. Garroni, A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182 (2006) 125-152. | MR 2247954 | Zbl 1098.74006

[21] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55-91. | MR 2244798 | Zbl 1101.74015

[22] J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms. II : Advanced theory and bundle methods, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 306. Springer-Verlag, Berlin (1993). | MR 1295240 | Zbl 0795.49002

[23] D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation. Math. Models Meth. Appl. Sci. 18 (2008) 1529-1569. | MR 2446401 | Zbl 1151.49014

[24] D. Knees, C. Zanini and A. Mielke, Crack propagation in polyconvex materials. Physica D 239 (2010) 1470-1484. | MR 2658341 | Zbl 1201.49013

[25] M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423-447. | Zbl 1133.74038

[26] P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in Nonlinear differential equations (Chvalatice, 1998), Res. Notes Math. 404, Chapman & Hall/CRC, Boca Raton, FL (1999) 47-110. | MR 1695378 | Zbl 0949.47053

[27] P. Krejčí, and M. Liero, Rate independent Kurzweil processes. Appl. Math. 54 (2009) 117-145. | MR 2491851 | Zbl 1212.49007

[28] C.J. Larsen, Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math. 63 (2010) 630-654. | MR 2583308 | Zbl pre05689532

[29] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems. Calc. Var. PDEs 22 (2005) 73-99. | MR 2105969 | Zbl 1161.74387

[30] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain. J. Nonlin. Sci. 19 (2009) 221-248. | MR 2511255 | Zbl 1173.49013

[31] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351-382. | MR 1999280 | Zbl 1068.74522

[32] A. Mielke, Evolution in rate-independent systems (Chap. 6), in Handbook of differential equations, evolutionary equations 2, C. Dafermos and E. Feireisl Eds., Elsevier B.V., Amsterdam (2005) 461-559. | MR 2182832 | Zbl 1120.47062

[33] A. Mielke, Differential, energetic and metric formulations for rate-independent processes. Lecture Notes, Summer School Cetraro (in press). | Zbl 1251.35003

[34] A. Mielke and T. Roubčíek, A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1 (2003) 571-597. | MR 2029592 | Zbl 1183.74207

[35] A. Mielke and T. Roubčíek, Rate-independent damage processes in nonlinear elasticity. M3 ! AS Math. Models Meth. Appl. Sci. 16 (2006) 177-209. | MR 2210087 | Zbl 1094.35068

[36] A. Mielke and T. Roubčíek, Rate-Independent Systems : Theory and Application. (In preparation).

[37] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R. Balean and R. Farwig Eds., Shaker-Verlag, Aachen (1999) 117-129.

[38] A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA 11 (2004) 151-189. | MR 2210284 | Zbl 1061.35182

[39] A. Mielke and A. Timofte, An energetic material model for time-dependent ferroelectric behavior : existence and uniqueness. Math. Meth. Appl. Sci. 29 (2006) 1393-1410. | MR 2247308 | Zbl 1096.74017

[40] A. Mielke and S. Zelik, On the vanishing viscosity limit in parabolic systems with rate-independent dissipation terms. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (submitted). | MR 3235058 | Zbl 1295.35036

[41] A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137-177. | MR 1897379 | Zbl 1012.74054

[42] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. 25 (2009) 585-615. | MR 2525194 | Zbl 1170.49036

[43] A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations. (In preparation). | Zbl 1270.35289

[44] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion. Math. Models Meth. Appl. Sci. 18 (2008) 1895-1925. | MR 2472402 | Zbl 1155.74035

[45] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). | MR 274683 | Zbl 0193.18401

[46] R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM : COCV 12 (2006) 564-614. | Numdam | MR 2224826 | Zbl 1116.34048

[47] R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications. Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008) 97-169. | Numdam | MR 2413674 | Zbl 1183.35164

[48] T. Roubčíek, Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32 (2009) 825-862. | MR 2507935 | Zbl 1194.35226

[49] U. Stefanelli, A variational characterization of rate-independent evolution. Math. Nachr. 282 (2009) 1492-1512. | MR 2573462 | Zbl 1217.34104

[50] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strains - Existence and regularity results. Zeits. Angew. Math. Mech. 90 (2009) 88-112. | MR 2640367 | Zbl 1191.35159

[51] R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth. Boll. Unione Mat. Ital. 2 (2009) 1-35. | MR 2493642 | Zbl 1180.35521

[52] A. Visintin, Differential models of hysteresis, Applied Mathematical Sciences 111. Springer-Verlag, Berlin (1994). | MR 1329094 | Zbl 0820.35004