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

We study a rate-independent system with non-convex energy and 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.

[1]  D. Knees.  Convergence analysis of time-discretisation schemes for rate-independent systems , 2017, ESAIM: Control, Optimisation and Calculus of Variations.

[2]  Alexander Mielke,et al.  Existence results for energetic models for rate-independent systems , 2005 .

[3]  E. Peterson,et al.  Generalized variational inequalities , 1982 .

[4]  M. Tvrdý.  Regulated functions and the Perron-Stieltjes integral , 1989 .

[5]  Riccarda Rossi,et al.  Variational Convergence of Gradient Flows and Rate-Independent Evolutions in Metric Spaces , 2012, 1207.2725.

[6]  V. Barbu,et al.  Convexity and optimization in banach spaces , 1972 .

[7]  Tomáš Roubíček,et al.  Rate-Independent Systems: Theory and Application , 2015 .

[8]  A. Mielke,et al.  BV SOLUTIONS AND VISCOSITY APPROXIMATIONS OF RATE-INDEPENDENT SYSTEMS ∗ , 2009, 0910.3360.

[9]  Alexander Mielke,et al.  On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity , 2006 .

[10]  J. Moreau.  Evolution problem associated with a moving convex set in a Hilbert space , 1977 .

[11]  F. Santambrogio,et al.  Sweeping processes with prescribed behavior on jumps , 2017, 1707.09765.

[12]  Riccarda Rossi,et al.  A metric approach to a class of doubly nonlinear evolution equations and applications , 2008 .

[13]  M. Liero,et al.  Rate independent Kurzweil processes , 2009 .

[14]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[15]  Vincenzo Recupero,et al.  BV solutions of rate independent variational inequalities , 2011 .

[16]  A. Mielke,et al.  On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms , 2014 .

[17]  Majid Fakhar,et al.  On generalized variational inequalities , 2009, J. Glob. Optim..

[18]  A. Mielke,et al.  Balanced Viscosity (BV) solutions to infinite-dimensional rate-independent systems , 2013, 1309.6291.

[19]  Vincenzo Recupero,et al.  Sweeping processes and rate independence , 2016 .

[20]  Pavel Krejčí,et al.  Comparing BV solutions of rate independent processes , 2014 .

[21]  G. Leoni.  A First Course in Sobolev Spaces , 2009 .

[22]  Riccarda Rossi,et al.  Modeling solutions with jumps for rate-independent systems on metric spaces , 2008, 0807.0744.