# American Institute of Mathematical Sciences

February  2013, 6(1): 167-191. doi: 10.3934/dcdss.2013.6.167

## A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems

 1 Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I-25133 Brescia, Italy 2 Dipartimento di Matematica “F. Casorati", Università di Pavia, Via Ferrata 1, I- 27100 Pavia, Italy

Received  September 2011 Revised  October 2011 Published  October 2012

The notion of BV solution to a rate-independent system was introduced in [8] to describe the vanishing viscosity limit (in the dissipation term) of doubly nonlinear evolution equations. Like energetic solutions [5] in the case of convex energies, BV solutions provide a careful description of rate-independent evolution driven by nonconvex energies, and in particular of the energetic behavior of the system at jumps.
In this paper we study both notions in the one-dimensional setting and we obtain a full characterization of BV and energetic solutions for a broad family of energy functionals. In the case of monotone loadings we provide a simple and explicit characterization of such solutions, which allows for a direct comparison of the two concepts.
Citation: Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'', Oxford Mathematical Monographs, (2000). Google Scholar [2] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity,, J. Convex Analysis, 13 (2006), 151. Google Scholar [3] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnels,'', Dunod, (1974). Google Scholar [4] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73. doi: 10.1007/s00526-004-0267-8. Google Scholar [5] A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461. Google Scholar [6] A. Mielke, Differential, energetic and metric formulations for rate-independent processes,, in, (2008), 87. Google Scholar [7] 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. doi: 10.3934/dcds.2009.25.585. Google Scholar [8] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems,, ESAIM Control Optim. Calc. Var, 18 (2012), 36. doi: 10.1051/cocv/2010054.Published. Google Scholar [9] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, in, (1999), 117. Google Scholar [10] A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151. doi: 10.1007/s00030-003-1052-7. Google Scholar [11] L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system,, SIAM J. Math. Anal., 41 (2009), 1340. doi: 10.1137/090750809. Google Scholar [12] M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation,, Adv. Calc. Var., 3 (2010), 149. Google Scholar [13] U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492. doi: 10.1002/mana.200810803. Google Scholar [14] A. Visintin, "Differential Models of Hysteresis,'', Applied Mathematical Sciences, (1994). Google Scholar

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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'', Oxford Mathematical Monographs, (2000). Google Scholar [2] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity,, J. Convex Analysis, 13 (2006), 151. Google Scholar [3] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnels,'', Dunod, (1974). Google Scholar [4] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73. doi: 10.1007/s00526-004-0267-8. Google Scholar [5] A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461. Google Scholar [6] A. Mielke, Differential, energetic and metric formulations for rate-independent processes,, in, (2008), 87. Google Scholar [7] 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. doi: 10.3934/dcds.2009.25.585. Google Scholar [8] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems,, ESAIM Control Optim. Calc. Var, 18 (2012), 36. doi: 10.1051/cocv/2010054.Published. Google Scholar [9] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, in, (1999), 117. Google Scholar [10] A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151. doi: 10.1007/s00030-003-1052-7. Google Scholar [11] L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system,, SIAM J. Math. Anal., 41 (2009), 1340. doi: 10.1137/090750809. Google Scholar [12] M. Negri, A comparative analysis on variational models for quasi-static brittle crack propagation,, Adv. Calc. Var., 3 (2010), 149. Google Scholar [13] U. Stefanelli, A variational characterization of rate-independent evolution,, Math. Nachr., 282 (2009), 1492. doi: 10.1002/mana.200810803. Google Scholar [14] A. Visintin, "Differential Models of Hysteresis,'', Applied Mathematical Sciences, (1994). Google Scholar
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