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Rate-independent evolution of sets
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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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