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Hysteresis operators in metric spaces
P.D.E.s with hysteresis 30 years later
1. | Dipartimento di Matematica dell'Università degli Studi di Trento, via Sommarive 14, 38123 Povo di Trento, Italy |
  Two examples of initial- and boundary-value problems for P.D.E.s with hysteresis are then illustrated. Well-posedness is proved for quasilinear parabolic problems with either continuous or discontinuous hysteresis. Existence of a weak solution is shown for second-order quasilinear hyperbolic problems with discontinuous hysteresis.
References:
[1] |
G. Bertotti, Hysteresis in Magnetism, Academic Press, Boston, 1998. |
[2] |
G. Bertotti and I. Mayergoyz, eds., The Science of Hysteresis, Elsevier, Oxford, 2006. |
[3] |
R. Bouc, Solution périodique de l'équation de la ferrorésonance avec hystérésis, C.R. Acad. Sci. Paris, Série A, 263 (1966), A497-A499. |
[4] |
R. Bouc, Modèle Mathématique D'hystérésis et Application Aux Systèmes à un Degré de Liberté, Thèse, Marseille, 1969. |
[5] |
M. Brokate, On a characterization of the Preisach model for hysteresis, Rend. Sem. Mat. Padova, 83 (1990), 153-163. |
[6] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Heidelberg, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[7] |
M. Brokate and A. Visintin, Properties of the Preisach model for hysteresis, J. Reine Angew. Math., 402 (1989), 1-40.
doi: 10.1515/crll.1989.402.1. |
[8] |
G. Dal Maso and R. Toader, A model for the quasi-static growth or brittle fractures: Existence and approximation results, Arch. Rat. Mech. Anal., 162 (2002), 101-135.
doi: 10.1007/s002050100187. |
[9] |
A. Damlamian and A. Visintin, Une généralisation vectorielle du modèle de Preisach pour l'hystérésis, C. R. Acad. Sci. Paris, Série I, 297 (1983), 437-440. |
[10] |
E. Della Torre, Magnetic Hysteresis, Wiley and I.E.E.E. Press, 1999. |
[11] |
P. Duhem, The Evolution of Mechanics, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980. Original edition: L'évolution de la méchanique, Joanin, Paris, 1903. |
[12] |
G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem: Existence and approximation results, J. Mech. Phys. Solids, 46 (1998), 1319-1342.
doi: 10.1016/S0022-5096(98)00034-9. |
[13] |
M. Hilpert, On uniqueness for evolution problems with hysteresis, in Mathematical Models for Phase Change Problems (ed. J.-F. Rodrigues), Internat. Ser. Numer. Math., 88, Birkhäuser, Basel, 1989, 377-388. |
[14] |
D. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman and Hall, London, 1991. |
[15] |
M. A. Krasnosel'skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabreĭko, E. A. Lifsic and A. V. Pokrovskiĭ, Hysterant operator, Soviet Math. Dokl., 11 (1970), 29-33. |
[16] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Berlin, 1989. Russian edition: Nauka, Moscow, 1983.
doi: 10.1007/978-3-642-61302-9. |
[17] |
P. Krejčí, Hysteresis and periodic solutions of semi-linear and quasi-linear wave equations, Math. Z., 193 (1986), 247-264.
doi: 10.1007/BF01174335. |
[18] |
P. Krejčí, Convexity, Hysteresis and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996. |
[19] |
I. D. Mayergoyz, Mathematical models of hysteresis, Phys. Rev. Letters, 56 (1986), 1518-1521.
doi: 10.1103/PhysRevLett.56.1518. |
[20] |
I. D. Mayergoyz, Mathematical models of hysteresis, I.E.E.E. Trans. Magn., 22 (1986), 603-608. |
[21] |
I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991.
doi: 10.2172/6911694. |
[22] |
I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, Amsterdam, 2003. |
[23] |
A. Mielke, Evolution of rate-independent systems, in Evolutionary equations. Vol. II (eds. C. Dafermos and E. Feireisel), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 461-559. |
[24] |
A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. |
[25] |
A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177.
doi: 10.1007/s002050200194. |
[26] |
F. Preisach, Über die magnetische Nachwirkung, Z. Physik, 94 (1935), 277-302. |
[27] |
A. Visintin, Hystérésis dans les systèmes distribués, C.R. Acad. Sci. Paris, Série I, 293 (1981), 625-628. |
[28] |
A. Visintin, A model for hysteresis of distributed systems, Ann. Mat. Pura Appl., 131 (1982), 203-231.
doi: 10.1007/BF01765153. |
[29] |
A. Visintin, A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5-18. |
[30] |
A. Visintin, Continuity properties of a class of hysteresis functionals, Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 232-247. |
[31] |
A. Visintin, Hysteresis and semigroups, in Models of Hysteresis (ed. A. Visintin), Pitman Res. Notes Math. Ser., 286, Longman, Harlow, 1993, 192-206. |
[32] |
A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994.
doi: 10.1007/978-3-662-11557-2. |
[33] |
A. Visintin, Quasi-linear hyperbolic equations with hysteresis, Ann. Inst. H. Poincaré. Analyse Non Linéaire, 19 (2002), 451-476.
doi: 10.1016/S0294-1449(01)00086-5. |
[34] |
A. Visintin, Maxwell's equations with vector hysteresis, Arch. Rat. Mech. Anal., 175 (2005), 1-37.
doi: 10.1007/s00205-004-0333-6. |
[35] |
A. Visintin, Mathematical models of hysteresis, in Modelling and Optimization of Distributed Parameter Systems (Warsaw, 1995), Chapman & Hall, New York, 1996, 71-80. |
[36] |
A. Visintin, Rheological models vs. homogenization, G.A.M.M.-Mitt., 34 (2011), 113-117.
doi: 10.1002/gamm.201110018. |
show all references
References:
[1] |
G. Bertotti, Hysteresis in Magnetism, Academic Press, Boston, 1998. |
[2] |
G. Bertotti and I. Mayergoyz, eds., The Science of Hysteresis, Elsevier, Oxford, 2006. |
[3] |
R. Bouc, Solution périodique de l'équation de la ferrorésonance avec hystérésis, C.R. Acad. Sci. Paris, Série A, 263 (1966), A497-A499. |
[4] |
R. Bouc, Modèle Mathématique D'hystérésis et Application Aux Systèmes à un Degré de Liberté, Thèse, Marseille, 1969. |
[5] |
M. Brokate, On a characterization of the Preisach model for hysteresis, Rend. Sem. Mat. Padova, 83 (1990), 153-163. |
[6] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Heidelberg, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[7] |
M. Brokate and A. Visintin, Properties of the Preisach model for hysteresis, J. Reine Angew. Math., 402 (1989), 1-40.
doi: 10.1515/crll.1989.402.1. |
[8] |
G. Dal Maso and R. Toader, A model for the quasi-static growth or brittle fractures: Existence and approximation results, Arch. Rat. Mech. Anal., 162 (2002), 101-135.
doi: 10.1007/s002050100187. |
[9] |
A. Damlamian and A. Visintin, Une généralisation vectorielle du modèle de Preisach pour l'hystérésis, C. R. Acad. Sci. Paris, Série I, 297 (1983), 437-440. |
[10] |
E. Della Torre, Magnetic Hysteresis, Wiley and I.E.E.E. Press, 1999. |
[11] |
P. Duhem, The Evolution of Mechanics, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980. Original edition: L'évolution de la méchanique, Joanin, Paris, 1903. |
[12] |
G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem: Existence and approximation results, J. Mech. Phys. Solids, 46 (1998), 1319-1342.
doi: 10.1016/S0022-5096(98)00034-9. |
[13] |
M. Hilpert, On uniqueness for evolution problems with hysteresis, in Mathematical Models for Phase Change Problems (ed. J.-F. Rodrigues), Internat. Ser. Numer. Math., 88, Birkhäuser, Basel, 1989, 377-388. |
[14] |
D. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman and Hall, London, 1991. |
[15] |
M. A. Krasnosel'skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabreĭko, E. A. Lifsic and A. V. Pokrovskiĭ, Hysterant operator, Soviet Math. Dokl., 11 (1970), 29-33. |
[16] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Berlin, 1989. Russian edition: Nauka, Moscow, 1983.
doi: 10.1007/978-3-642-61302-9. |
[17] |
P. Krejčí, Hysteresis and periodic solutions of semi-linear and quasi-linear wave equations, Math. Z., 193 (1986), 247-264.
doi: 10.1007/BF01174335. |
[18] |
P. Krejčí, Convexity, Hysteresis and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996. |
[19] |
I. D. Mayergoyz, Mathematical models of hysteresis, Phys. Rev. Letters, 56 (1986), 1518-1521.
doi: 10.1103/PhysRevLett.56.1518. |
[20] |
I. D. Mayergoyz, Mathematical models of hysteresis, I.E.E.E. Trans. Magn., 22 (1986), 603-608. |
[21] |
I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991.
doi: 10.2172/6911694. |
[22] |
I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, Amsterdam, 2003. |
[23] |
A. Mielke, Evolution of rate-independent systems, in Evolutionary equations. Vol. II (eds. C. Dafermos and E. Feireisel), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 461-559. |
[24] |
A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. |
[25] |
A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177.
doi: 10.1007/s002050200194. |
[26] |
F. Preisach, Über die magnetische Nachwirkung, Z. Physik, 94 (1935), 277-302. |
[27] |
A. Visintin, Hystérésis dans les systèmes distribués, C.R. Acad. Sci. Paris, Série I, 293 (1981), 625-628. |
[28] |
A. Visintin, A model for hysteresis of distributed systems, Ann. Mat. Pura Appl., 131 (1982), 203-231.
doi: 10.1007/BF01765153. |
[29] |
A. Visintin, A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5-18. |
[30] |
A. Visintin, Continuity properties of a class of hysteresis functionals, Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 232-247. |
[31] |
A. Visintin, Hysteresis and semigroups, in Models of Hysteresis (ed. A. Visintin), Pitman Res. Notes Math. Ser., 286, Longman, Harlow, 1993, 192-206. |
[32] |
A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994.
doi: 10.1007/978-3-662-11557-2. |
[33] |
A. Visintin, Quasi-linear hyperbolic equations with hysteresis, Ann. Inst. H. Poincaré. Analyse Non Linéaire, 19 (2002), 451-476.
doi: 10.1016/S0294-1449(01)00086-5. |
[34] |
A. Visintin, Maxwell's equations with vector hysteresis, Arch. Rat. Mech. Anal., 175 (2005), 1-37.
doi: 10.1007/s00205-004-0333-6. |
[35] |
A. Visintin, Mathematical models of hysteresis, in Modelling and Optimization of Distributed Parameter Systems (Warsaw, 1995), Chapman & Hall, New York, 1996, 71-80. |
[36] |
A. Visintin, Rheological models vs. homogenization, G.A.M.M.-Mitt., 34 (2011), 113-117.
doi: 10.1002/gamm.201110018. |
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