American Institute of Mathematical Sciences

Advanced
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  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 & Continuous Dynamical Systems - S, 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

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.  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

[1]

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
[2]

Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070
[3]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020304
[4]

Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i
[5]

T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215
[6]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
[7]

Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591
[8]

Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256
[9]

Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure & Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479
[10]

Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101
[11]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076
[12]

Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[13]

Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585
[14]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
[15]

Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i
[16]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[17]

Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050
[18]

Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257
[19]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411
[20]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (29)
  • HTML views (196)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences