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.

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

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

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

, and ,[4] 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

, and ,[5] 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

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

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

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

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

and ,[10] 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

and ,[11] 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

and ,[12] 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

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

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

, and ,[15] Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media 3 (2008) 567-614. | MR 2425074 | Zbl 1156.74308

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

, , and ,[17] 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

, and ,[18] 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

and ,[19] 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] A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182 (2006) 125-152. | MR 2247954 | Zbl 1098.74006

and ,[21] 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

and ,[22] 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

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

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

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

, and ,[26] 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] Rate independent Kurzweil processes. Appl. Math. 54 (2009) 117-145. | MR 2491851 | Zbl 1212.49007

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

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

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

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

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

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

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

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

and ,[37] 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.

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

and ,[39] 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

and ,[40] 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

and ,[41] 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

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

, and ,[43] Nonsmooth analysis of doubly nonlinear evolution equations. (In preparation). | Zbl 1270.35289

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

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

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

and ,[47] 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

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

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

,[50] 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

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

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

,