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.
Mots-clés : 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},
pages = {36--80},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010054},
mrnumber = {2887927},
zbl = {1250.49041},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010054/}
}
TY - JOUR AU - Mielke, Alexander AU - Rossi, Riccarda AU - Savaré, Giuseppe TI - BV solutions and viscosity approximations of rate-independent systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 36 EP - 80 VL - 18 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010054/ DO - 10.1051/cocv/2010054 LA - en ID - COCV_2012__18_1_36_0 ER -
%0 Journal Article %A Mielke, Alexander %A Rossi, Riccarda %A Savaré, Giuseppe %T BV solutions and viscosity approximations of rate-independent systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 36-80 %V 18 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010054/ %R 10.1051/cocv/2010054 %G en %F 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, Tome 18 (2012) no. 1, pp. 36-80. doi : 10.1051/cocv/2010054. http://www.numdam.org/articles/10.1051/cocv/2010054/
[1] , Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR | Zbl
[2] and , A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108 (1990) 691-702. | MR | Zbl
[3] , and , Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000). | MR | Zbl
[4] , and , Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, 2nd edn., Birkhäuser Verlag, Basel (2008). | MR | Zbl
[5] , and , A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Meth. Appl. Sci. 18 (2008) 125-164. | MR | Zbl
[6] , and , A complete-damage problem at small strains. Z. Angew. Math. Phys. 60 (2009) 205-236. | MR | Zbl
[7] , and , Existence and construction of bipotentials for graphs of multivalued laws. J. Convex Anal. 15 (2008) 87-104. | MR | Zbl
[8] , On some doubly nonlinear evolution equations in Banach spaces. Japan J. Indust. Appl. Math. 9 (1992) 181-203. | MR | Zbl
[9] and , On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15 (1990) 737-756. | MR | Zbl
[10] and , A model for quasi-static growth of brittle fractures : existence and approximation results. Arch. Ration. Mech. Anal. 162 (2002) 101-135. | MR | Zbl
[11] and , A model for the quasi-static growth of brittle fractures based on local minimization. Math. Models Meth. Appl. Sci. 12 (2002) 1773-1799. | MR | Zbl
[12] and , 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 | Zbl
[13] , and , Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176 (2005) 165-225. | MR | Zbl
[14] , and , Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237-291. | MR | Zbl
[15] , , and , Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media 3 (2008) 567-614. | MR | Zbl
[16] , , and , A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189 (2008) 469-544. | MR | Zbl
[17] , and , Quasistatic evolution for cam-clay plasiticity : a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differential Equations (to appear). | MR
[18] and , On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Analysis 13 (2006) 151-167. | MR | Zbl
[19] , A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy. Ann. Inst. Henri Poincaré, Anal. Non Linéaire (to appear). | Numdam | Zbl
[20] and , A variational view of partial brittle damage evolution. Arch. Ration. Mech. Anal. 182 (2006) 125-152. | MR | Zbl
[21] and , Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595 (2006) 55-91. | MR | Zbl
[22] and , 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 | Zbl
[23] , and , On the inviscid limit of a model for crack propagation. Math. Models Meth. Appl. Sci. 18 (2008) 1529-1569. | MR | Zbl
[24] , and , Crack propagation in polyconvex materials. Physica D 239 (2010) 1470-1484. | MR | Zbl
[25] , and , A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423-447. | Zbl
[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 | Zbl
[27] , and , Rate independent Kurzweil processes. Appl. Math. 54 (2009) 117-145. | MR | Zbl
[28] , Epsilon-stable quasi-static brittle fracture evolution. Comm. Pure Appl. Math. 63 (2010) 630-654. | MR
[29] and , Existence results for energetic models for rate-independent systems. Calc. Var. PDEs 22 (2005) 73-99. | MR | Zbl
[30] and , Global existence for rate-independent gradient plasticity at finite strain. J. Nonlin. Sci. 19 (2009) 221-248. | MR | Zbl
[31] , Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15 (2003) 351-382. | MR | Zbl
[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 | Zbl
[33] , Differential, energetic and metric formulations for rate-independent processes. Lecture Notes, Summer School Cetraro (in press). | Zbl
[34] and , A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1 (2003) 571-597. | MR | Zbl
[35] and , Rate-independent damage processes in nonlinear elasticity. M3 ! AS Math. Models Meth. Appl. Sci. 16 (2006) 177-209. | MR | Zbl
[36] and , Rate-Independent Systems : Theory and Application. (In preparation).
[37] and , 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] and , On rate-independent hysteresis models. NoDEA 11 (2004) 151-189. | MR | Zbl
[39] and , An energetic material model for time-dependent ferroelectric behavior : existence and uniqueness. Math. Meth. Appl. Sci. 29 (2006) 1393-1410. | MR | Zbl
[40] and , On the vanishing viscosity limit in parabolic systems with rate-independent dissipation terms. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (submitted). | MR | Zbl
[41] , and , A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137-177. | MR | Zbl
[42] , and , Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. 25 (2009) 585-615. | MR | Zbl
[43] , and , Nonsmooth analysis of doubly nonlinear evolution equations. (In preparation). | Zbl
[44] and , Quasi-static crack propagation by Griffith's criterion. Math. Models Meth. Appl. Sci. 18 (2008) 1895-1925. | MR | Zbl
[45] , Convex Analysis. Princeton University Press, Princeton (1970). | MR | Zbl
[46] and , Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM : COCV 12 (2006) 564-614. | Numdam | MR | Zbl
[47] , and , 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 | Zbl
[48] , Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32 (2009) 825-862. | MR | Zbl
[49] , A variational characterization of rate-independent evolution. Math. Nachr. 282 (2009) 1492-1512. | MR | Zbl
[50] and , Damage of nonlinearly elastic materials at small strains - Existence and regularity results. Zeits. Angew. Math. Mech. 90 (2009) 88-112. | MR | Zbl
[51] and , An artificial viscosity approach to quasistatic crack growth. Boll. Unione Mat. Ital. 2 (2009) 1-35. | MR | Zbl
[52] , Differential models of hysteresis, Applied Mathematical Sciences 111. Springer-Verlag, Berlin (1994). | MR | Zbl
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