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Thermalization of rate-independent processes by entropic regularization
1. | Applied & Computational Mathematics and Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125-9400, United States |
2. | School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088, United States |
3. | Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom |
4. | Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, United States |
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, second ed., 2008. |
[2] |
S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004. |
[3] |
R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200. |
[4] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. |
[5] |
M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003. |
[6] |
A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. |
[7] |
\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf. |
[8] |
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. |
[9] |
A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. |
[10] |
J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299. |
[11] |
J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166. |
[12] |
D. Preiss, Geometry of measures in $\mathbbR^n$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643. |
[13] |
T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009. |
[14] |
T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077. |
[15] |
K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965. |
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, second ed., 2008. |
[2] |
S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004. |
[3] |
R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200. |
[4] |
R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. |
[5] |
M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003. |
[6] |
A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. |
[7] |
\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf. |
[8] |
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. |
[9] |
A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. |
[10] |
J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299. |
[11] |
J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166. |
[12] |
D. Preiss, Geometry of measures in $\mathbbR^n$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643. |
[13] |
T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009. |
[14] |
T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077. |
[15] |
K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965. |
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