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

P.D.E.s with hysteresis 30 years later

Augusto Visintin 1,

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.
Keywords: quasilinear hyperbolic partial differential equations., hysteresis operators, quasilinear parabolic partial differential equations, continuous and discontinuous hysteresis, Hysteresis.
Mathematics Subject Classification: 35K60, 35K59, 35L72, 47J4.
Citation: Augusto Visintin. P.D.E.s with hysteresis 30 years later. Discrete & 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. Google Scholar

[2]

G. Bertotti and I. Mayergoyz, eds., The Science of Hysteresis, Elsevier, Oxford, 2006. Google Scholar

[3]

R. Bouc, Solution périodique de l'équation de la ferrorésonance avec hystérésis, C.R. Acad. Sci. Paris, Série A, 263 (1966), A497-A499.  Google Scholar

[4]

R. Bouc, Modèle Mathématique D'hystérésis et Application Aux Systèmes à un Degré de Liberté, Thèse, Marseille, 1969. Google Scholar

[5]

M. Brokate, On a characterization of the Preisach model for hysteresis, Rend. Sem. Mat. Padova, 83 (1990), 153-163.  Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Heidelberg, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Della Torre, Magnetic Hysteresis, Wiley and I.E.E.E. Press, 1999. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman and Hall, London, 1991. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. Krejčí, Convexity, Hysteresis and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.  Google Scholar

[19]

I. D. Mayergoyz, Mathematical models of hysteresis, Phys. Rev. Letters, 56 (1986), 1518-1521. doi: 10.1103/PhysRevLett.56.1518.  Google Scholar

[20]

I. D. Mayergoyz, Mathematical models of hysteresis, I.E.E.E. Trans. Magn., 22 (1986), 603-608.  Google Scholar

[21]

I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. doi: 10.2172/6911694.  Google Scholar

[22]

I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, Amsterdam, 2003. Google Scholar

[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.  Google Scholar

[24]

A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. Google Scholar

[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.  Google Scholar

[26]

F. Preisach, Über die magnetische Nachwirkung, Z. Physik, 94 (1935), 277-302. Google Scholar

[27]

A. Visintin, Hystérésis dans les systèmes distribués, C.R. Acad. Sci. Paris, Série I, 293 (1981), 625-628.  Google Scholar

[28]

A. Visintin, A model for hysteresis of distributed systems, Ann. Mat. Pura Appl., 131 (1982), 203-231. doi: 10.1007/BF01765153.  Google Scholar

[29]

A. Visintin, A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5-18.  Google Scholar

[30]

A. Visintin, Continuity properties of a class of hysteresis functionals, Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 232-247.  Google Scholar

[31]

A. Visintin, Hysteresis and semigroups, in Models of Hysteresis (ed. A. Visintin), Pitman Res. Notes Math. Ser., 286, Longman, Harlow, 1993, 192-206.  Google Scholar

[32]

A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[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.  Google Scholar

[34]

A. Visintin, Maxwell's equations with vector hysteresis, Arch. Rat. Mech. Anal., 175 (2005), 1-37. doi: 10.1007/s00205-004-0333-6.  Google Scholar

[35]

A. Visintin, Mathematical models of hysteresis, in Modelling and Optimization of Distributed Parameter Systems (Warsaw, 1995), Chapman & Hall, New York, 1996, 71-80.  Google Scholar

[36]

A. Visintin, Rheological models vs. homogenization, G.A.M.M.-Mitt., 34 (2011), 113-117. doi: 10.1002/gamm.201110018.  Google Scholar

show all references

References:
[1]

G. Bertotti, Hysteresis in Magnetism, Academic Press, Boston, 1998. Google Scholar

[2]

G. Bertotti and I. Mayergoyz, eds., The Science of Hysteresis, Elsevier, Oxford, 2006. Google Scholar

[3]

R. Bouc, Solution périodique de l'équation de la ferrorésonance avec hystérésis, C.R. Acad. Sci. Paris, Série A, 263 (1966), A497-A499.  Google Scholar

[4]

R. Bouc, Modèle Mathématique D'hystérésis et Application Aux Systèmes à un Degré de Liberté, Thèse, Marseille, 1969. Google Scholar

[5]

M. Brokate, On a characterization of the Preisach model for hysteresis, Rend. Sem. Mat. Padova, 83 (1990), 153-163.  Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, Heidelberg, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Della Torre, Magnetic Hysteresis, Wiley and I.E.E.E. Press, 1999. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Jiles, Introduction to Magnetism and Magnetic Materials, Chapman and Hall, London, 1991. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. Krejčí, Convexity, Hysteresis and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996.  Google Scholar

[19]

I. D. Mayergoyz, Mathematical models of hysteresis, Phys. Rev. Letters, 56 (1986), 1518-1521. doi: 10.1103/PhysRevLett.56.1518.  Google Scholar

[20]

I. D. Mayergoyz, Mathematical models of hysteresis, I.E.E.E. Trans. Magn., 22 (1986), 603-608.  Google Scholar

[21]

I. D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. doi: 10.2172/6911694.  Google Scholar

[22]

I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, Amsterdam, 2003. Google Scholar

[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.  Google Scholar

[24]

A. Mielke and T. Roubíček, Rate-Independent Systems - Theory and Application, Springer, New York, 2015. Google Scholar

[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.  Google Scholar

[26]

F. Preisach, Über die magnetische Nachwirkung, Z. Physik, 94 (1935), 277-302. Google Scholar

[27]

A. Visintin, Hystérésis dans les systèmes distribués, C.R. Acad. Sci. Paris, Série I, 293 (1981), 625-628.  Google Scholar

[28]

A. Visintin, A model for hysteresis of distributed systems, Ann. Mat. Pura Appl., 131 (1982), 203-231. doi: 10.1007/BF01765153.  Google Scholar

[29]

A. Visintin, A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5-18.  Google Scholar

[30]

A. Visintin, Continuity properties of a class of hysteresis functionals, Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 232-247.  Google Scholar

[31]

A. Visintin, Hysteresis and semigroups, in Models of Hysteresis (ed. A. Visintin), Pitman Res. Notes Math. Ser., 286, Longman, Harlow, 1993, 192-206.  Google Scholar

[32]

A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.  Google Scholar

[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.  Google Scholar

[34]

A. Visintin, Maxwell's equations with vector hysteresis, Arch. Rat. Mech. Anal., 175 (2005), 1-37. doi: 10.1007/s00205-004-0333-6.  Google Scholar

[35]

A. Visintin, Mathematical models of hysteresis, in Modelling and Optimization of Distributed Parameter Systems (Warsaw, 1995), Chapman & Hall, New York, 1996, 71-80.  Google Scholar

[36]

A. Visintin, Rheological models vs. homogenization, G.A.M.M.-Mitt., 34 (2011), 113-117. doi: 10.1002/gamm.201110018.  Google Scholar

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