# 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, The Clarendon Press Oxford University Press, New York, 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), 151-167.  Google Scholar [3] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnels,'' Dunod, Gauthier-Villars, Paris, 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), 73-99. doi: 10.1007/s00526-004-0267-8.  Google Scholar [5] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam II (2005), 461-559.  Google Scholar [6] A. Mielke, Differential, energetic and metric formulations for rate-independent processes, in "Nonlinear PDE's and Applications'' (eds: L. Ambrosio and G. Savaré), Lecture Notes, C. I. M. E. Summer School, Cetraro, Italy, (2008), 87-171, Springer, 2011.  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), 585-615. 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, Calc., 18 (2012), 36-80. 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 "Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering'' (eds. H.-D. Alber, R. M. Balean and R. Farwig), Aachen, Shaker-Verlag, (1999), 117-129. Google Scholar [10] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. 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), 1340-1365. 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), 149-212.  Google Scholar [13] U. Stefanelli, A variational characterization of rate-independent evolution, Math. Nachr., 282 (2009), 1492-1512. doi: 10.1002/mana.200810803.  Google Scholar [14] A. Visintin, "Differential Models of Hysteresis,'' Applied Mathematical Sciences, 111, Springer-Verlag, Berlin, 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, The Clarendon Press Oxford University Press, New York, 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), 151-167.  Google Scholar [3] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnels,'' Dunod, Gauthier-Villars, Paris, 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), 73-99. doi: 10.1007/s00526-004-0267-8.  Google Scholar [5] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam II (2005), 461-559.  Google Scholar [6] A. Mielke, Differential, energetic and metric formulations for rate-independent processes, in "Nonlinear PDE's and Applications'' (eds: L. Ambrosio and G. Savaré), Lecture Notes, C. I. M. E. Summer School, Cetraro, Italy, (2008), 87-171, Springer, 2011.  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), 585-615. 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, Calc., 18 (2012), 36-80. 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 "Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering'' (eds. H.-D. Alber, R. M. Balean and R. Farwig), Aachen, Shaker-Verlag, (1999), 117-129. Google Scholar [10] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. 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), 1340-1365. 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), 149-212.  Google Scholar [13] U. Stefanelli, A variational characterization of rate-independent evolution, Math. Nachr., 282 (2009), 1492-1512. doi: 10.1002/mana.200810803.  Google Scholar [14] A. Visintin, "Differential Models of Hysteresis,'' Applied Mathematical Sciences, 111, Springer-Verlag, Berlin, 1994.  Google Scholar
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