August  2015, 8(4): 793-816. doi: 10.3934/dcdss.2015.8.793

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

Received  September 2013 Revised  July 2014 Published  October 2014

Continuous and discontinuous hysteresis operators are first reviewed in general. The Duhem model, the generalized play, the (delayed) relay and the Preisach model are outlined, as well as vector extensions of the two latter models.
    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.
Citation: Augusto Visintin. P.D.E.s with hysteresis 30 years later. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 793-816. doi: 10.3934/dcdss.2015.8.793
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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