# American Institute of Mathematical Sciences

January  2021, 14(1): 121-149. doi: 10.3934/dcdss.2020332

## Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads

 1 University of Kassel, Institute for Mathematics, Heinrich-Plett-Str. 40, 34132 Kassel, Germany 2 Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

* Corresponding author: Dorothee Knees

Dedicated to Alexander Mielke to the occasion of his 60th birthday

Received  September 2019 Revised  November 2019 Published  April 2020

We study a rate-independent system with non-convex energy in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized $BV$-solutions is obtained via vanishing viscosity in a suitable parameterized setting. In addition, we prove that the solution set is compact.

Citation: Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.  Google Scholar [2] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Mathematics and its Applications, 10. D. Reidel Publishing Co., Dordrecht, Editura Academiei Republicii Socialiste Romania, Bucharest, 1986.  Google Scholar [3] J. Dieudonné, Foundations of Modern Analysis. Enlarged and Corrected Printing, Pure and Applied Mathematics, Vol. 10-Ⅰ. Academic Press, New York-London, 1969.  Google Scholar [4] M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167.   Google Scholar [5] P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.   Google Scholar [6] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5.  Google Scholar [7] D. Knees, Convergence analysis of time-discretization schemes for rate-independent systems, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 65, 38 pp. doi: 10.1051/cocv/2018048.  Google Scholar [8] P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21 (2014), 121-146.   Google Scholar [9] D. Knees and S. Thomas, Optimal Control of a Rate-Independent System Constrained to Parametrized Balanced Viscosity Solutions, University of Kassel, 2018, arXiv: 1810.12572. Google Scholar [10] G. Leoni, A First Course in Sobolev Spaces, Second edition, Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017.  Google Scholar [11] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differ. Equ., 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar [12] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar [13] A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar [14] 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 [15] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar [16] A. Mielke, R. Rossi and G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan J. Math., 80 (2012), 381-410.  doi: 10.1007/s00032-012-0190-y.  Google Scholar [17] A. Mielke, R. Rossi and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc. (JEMS), 18 (2016), 2107-2165.  doi: 10.4171/JEMS/639.  Google Scholar [18] A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 67-135.   Google Scholar [19] V. Recupero, $BV$solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 269-315.   Google Scholar [20] V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23 (2016), 921-946.   Google Scholar [21] 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.   Google Scholar [22] V. Recupero and F. Santambrogio, Sweeping processes with prescribed behavior on jumps, Ann. Mat. Pura Appl., 197 (2018), 1311-1332.  doi: 10.1007/s10231-018-0726-z.  Google Scholar [23] M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pěst. Mat., 114 (1989), 187–209.  Google Scholar

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##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.  Google Scholar [2] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Mathematics and its Applications, 10. D. Reidel Publishing Co., Dordrecht, Editura Academiei Republicii Socialiste Romania, Bucharest, 1986.  Google Scholar [3] J. Dieudonné, Foundations of Modern Analysis. Enlarged and Corrected Printing, Pure and Applied Mathematics, Vol. 10-Ⅰ. Academic Press, New York-London, 1969.  Google Scholar [4] M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), 151-167.   Google Scholar [5] P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.   Google Scholar [6] P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145.  doi: 10.1007/s10492-009-0009-5.  Google Scholar [7] D. Knees, Convergence analysis of time-discretization schemes for rate-independent systems, ESAIM Control Optim. Calc. Var., 25 (2019), Art. 65, 38 pp. doi: 10.1051/cocv/2018048.  Google Scholar [8] P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21 (2014), 121-146.   Google Scholar [9] D. Knees and S. Thomas, Optimal Control of a Rate-Independent System Constrained to Parametrized Balanced Viscosity Solutions, University of Kassel, 2018, arXiv: 1810.12572. Google Scholar [10] G. Leoni, A First Course in Sobolev Spaces, Second edition, Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017.  Google Scholar [11] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differ. Equ., 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar [12] J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar [13] A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar [14] 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 [15] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar [16] A. Mielke, R. Rossi and G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan J. Math., 80 (2012), 381-410.  doi: 10.1007/s00032-012-0190-y.  Google Scholar [17] A. Mielke, R. Rossi and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc. (JEMS), 18 (2016), 2107-2165.  doi: 10.4171/JEMS/639.  Google Scholar [18] A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (2014), 67-135.   Google Scholar [19] V. Recupero, $BV$solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 269-315.   Google Scholar [20] V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23 (2016), 921-946.   Google Scholar [21] 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.   Google Scholar [22] V. Recupero and F. Santambrogio, Sweeping processes with prescribed behavior on jumps, Ann. Mat. Pura Appl., 197 (2018), 1311-1332.  doi: 10.1007/s10231-018-0726-z.  Google Scholar [23] M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pěst. Mat., 114 (1989), 187–209.  Google Scholar
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