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

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

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

Dorothee Knees ^1,, and Chiara Zanini ^2,

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 January 2021 Early access 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.

Keywords: Rate-independent system, discontinuous load, parameterized BV-solution, time-incremental minimum problems, vanishing viscosity limit.

Mathematics Subject Classification: Primary: 35R05, 49J40; Secondary: 74C05, 35Q74, 35D40, 49J45.

Citation: Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete and 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.
[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.
[3]	J. Dieudonné, Foundations of Modern Analysis. Enlarged and Corrected Printing, Pure and Applied Mathematics, Vol. 10-Ⅰ. Academic Press, New York-London, 1969.
[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.
[5]	P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[6]	P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.
[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.
[8]	P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21 (2014), 121-146.
[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.
[10]	G. Leoni, A First Course in Sobolev Spaces, Second edition, Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	V. Recupero, $BV$solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 269-315.
[20]	V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23 (2016), 921-946.
[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.
[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.
[23]	M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pěst. Mat., 114 (1989), 187–209.

show all references

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.
[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.
[3]	J. Dieudonné, Foundations of Modern Analysis. Enlarged and Corrected Printing, Pure and Applied Mathematics, Vol. 10-Ⅰ. Academic Press, New York-London, 1969.
[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.
[5]	P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.
[6]	P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.
[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.
[8]	P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21 (2014), 121-146.
[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.
[10]	G. Leoni, A First Course in Sobolev Spaces, Second edition, Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	V. Recupero, $BV$solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 269-315.
[20]	V. Recupero, Sweeping processes and rate independence, J. Convex Anal., 23 (2016), 921-946.
[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.
[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.
[23]	M. Tvrdý, Regulated functions and the Perron-Stieltjes integral, Časopis Pěst. Mat., 114 (1989), 187–209.

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