
-
Previous Article
Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network
- DCDS Home
- This Issue
-
Next Article
Hitting times distribution and extreme value laws for semi-flows
Visco-Energetic solutions to one-dimensional rate-independent problems
Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy |
Visco-Energetic solutions of rate-independent systems (recently introduced in [
In the present paper we study Visco-Energetic solutions in the scalar-valued case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction $δ$.
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. |
[2] |
G. Dal Maso and R. Toader,
A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799.
doi: 10.1142/S0218202502002331. |
[3] |
M. Efendiev and A. Mielke,
On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006), 151-167.
|
[4] |
I. S. Gál,
On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957), 309-320.
doi: 10.1090/S0002-9947-1957-0093562-7. |
[5] |
D. Knees and A. Schröder,
Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013), 63-99.
doi: 10.3934/dcdss.2013.6.63. |
[6] |
K. Kuratowski,
Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67.
doi: 10.1007/BF02416522. |
[7] |
D. Leguillon,
Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002), 61-72.
doi: 10.1016/S0997-7538(01)01184-6. |
[8] |
A. Mainik and A. Mielke,
Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[9] |
A. Mielke,
Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011), 87-170.
doi: 10.1007/978-3-642-21861-3_3. |
[10] |
A. Mielke, R. Rossi and G. Savaré,
Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009), 585-615.
doi: 10.3934/dcds.2009.25.585. |
[11] |
Alexander Mielke, Riccarda Rossi and Giuseppe Savaré,
BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.
doi: 10.1051/cocv/2010054. |
[12] |
Alexander Mielke, Riccarda Rossi and Giuseppe Savaré,
Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016), 2107-2165.
doi: 10.4171/JEMS/639. |
[13] |
A. Mielke and T. Roubíček,
Rate-Independent Systems: Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[14] |
A. Mielke and F. Theil,
On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
doi: 10.1007/s00030-003-1052-7. |
[15] |
A. Mielke, F. Theil and V. I. Levitas,
A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.
doi: 10.1007/s002050200194. |
[16] |
L. Minotti,
Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016. |
[17] |
L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60. |
[18] |
M. Negri and C. Ortner,
Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925.
doi: 10.1142/S0218202508003236. |
[19] |
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. (5), 7 (2008), 97-169.
|
[20] |
R. Rossi and G. Savaré,
A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 167-191.
|
[21] |
T. Roubíček, C. C. Panagiotopoulos and V. Mantic,
Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840.
doi: 10.1002/zamm.201200239. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000. |
[2] |
G. Dal Maso and R. Toader,
A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799.
doi: 10.1142/S0218202502002331. |
[3] |
M. Efendiev and A. Mielke,
On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006), 151-167.
|
[4] |
I. S. Gál,
On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957), 309-320.
doi: 10.1090/S0002-9947-1957-0093562-7. |
[5] |
D. Knees and A. Schröder,
Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013), 63-99.
doi: 10.3934/dcdss.2013.6.63. |
[6] |
K. Kuratowski,
Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67.
doi: 10.1007/BF02416522. |
[7] |
D. Leguillon,
Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002), 61-72.
doi: 10.1016/S0997-7538(01)01184-6. |
[8] |
A. Mainik and A. Mielke,
Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.
doi: 10.1007/s00526-004-0267-8. |
[9] |
A. Mielke,
Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011), 87-170.
doi: 10.1007/978-3-642-21861-3_3. |
[10] |
A. Mielke, R. Rossi and G. Savaré,
Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009), 585-615.
doi: 10.3934/dcds.2009.25.585. |
[11] |
Alexander Mielke, Riccarda Rossi and Giuseppe Savaré,
BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.
doi: 10.1051/cocv/2010054. |
[12] |
Alexander Mielke, Riccarda Rossi and Giuseppe Savaré,
Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016), 2107-2165.
doi: 10.4171/JEMS/639. |
[13] |
A. Mielke and T. Roubíček,
Rate-Independent Systems: Theory and Application, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2706-7. |
[14] |
A. Mielke and F. Theil,
On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
doi: 10.1007/s00030-003-1052-7. |
[15] |
A. Mielke, F. Theil and V. I. Levitas,
A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.
doi: 10.1007/s002050200194. |
[16] |
L. Minotti,
Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016. |
[17] |
L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60. |
[18] |
M. Negri and C. Ortner,
Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925.
doi: 10.1142/S0218202508003236. |
[19] |
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. (5), 7 (2008), 97-169.
|
[20] |
R. Rossi and G. Savaré,
A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 167-191.
|
[21] |
T. Roubíček, C. C. Panagiotopoulos and V. Mantic,
Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840.
doi: 10.1002/zamm.201200239. |







[1] |
Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167 |
[2] |
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 |
[3] |
Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585 |
[4] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[5] |
Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 |
[6] |
Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i |
[7] |
T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215 |
[8] |
Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257 |
[9] |
Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649 |
[10] |
Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 |
[11] |
Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145 |
[12] |
Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i |
[13] |
Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106 |
[14] |
Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028 |
[15] |
Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks and Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257 |
[16] |
Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591 |
[17] |
Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations and Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411 |
[18] |
Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029 |
[19] |
Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291 |
[20] |
Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]