February  2013, 6(1): 257-275. doi: 10.3934/dcdss.2013.6.257

Structural stability of rate-independent nonpotential flows

1. 

Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38050, Povo di Trento, Italy

Received  April 2011 Revised  August 2011 Published  October 2012

Several phenomena may be represented by doubly-nonlinear equations of the form $$ \alpha(D_tu) - \nabla\cdot \gamma(\nabla u)\ni h, $$ with $\alpha$ and $\gamma$ (possibly multivalued) maximal monotone mappings. Hysteresis effects are characterized by rate-independence, which corresponds to $\alpha$ positively homogeneous of zero degree.
    Fitzpatrick showed that any maximal monotone relation may be represented variationally. On this basis, an initial- and boundary-value problem associated to the equation above is here formulated as a null-minimization problem, without assuming $\gamma$ to be cyclically monotone. Existence of a solution $u\in H^1(0,T; H^1(\Omega))$ is proved, as well as its stability with respect to variations of the data, of the mapping $\gamma$, and of the domain $\Omega$.
Citation: Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
References:
[1]

S. Aizicovici and Q. Yan, Convergence theorems for abstract doubly nonlinear differential equations,, Panamer. Math. J, 7 (1997), 1.

[2]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitman, (1984).

[3]

G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations,, Differential Integral Equations, 6 (1993), 1161.

[4]

V. Barbu, Existence theorems for a class of two point boundary problems,, J. Differential Equations, 17 (1975), 236.

[5]

V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces,", Springer, (2010).

[6]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", Editura Academiei, (1978).

[7]

A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,", Oxford University Press, (1998).

[8]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (1973).

[9]

H. Brezis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983).

[10]

H. Brezis and I. Ekeland, Un principe variationnel associé àcertaines équations parabo-liques.I. Le cas indépendant du temps,, II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971.

[11]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). doi: 10.1007/978-1-4612-4048-8.

[12]

M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, J. Convex Anal., 15 (2008), 87.

[13]

R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements,, Set-Valued Analysis, 10 (2002), 297. doi: 10.1023/A:1020639314056.

[14]

R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product,, Proc. Amer. Math. Soc., 131 (2003), 2379. doi: 10.1090/S0002-9939-03-07053-9.

[15]

P. Colli, On some doubly nonlinear evolution equations in Banach spaces,, Japan J. Indust. Appl. Math., 9 (1992), 181. doi: 10.1007/BF03167565.

[16]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution problems,, Communications in P. D. E. s, 15 (1990), 737.

[17]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993).

[18]

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. doi: 10.1007/s002050100187.

[19]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842.

[20]

I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles,", Dunod Gau-thier-Villars, (1974).

[21]

W. Fenchel, "Convex Cones, Sets, and Functions,", Princeton Univ., (1953).

[22]

S. Fitzpatrick, Representing monotone operators by convex functions,, Workshop/Minicon-ference on Functional Analysis and Optimization (Canberra, (1988), 59.

[23]

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. doi: 10.1016/S0022-5096(98)00034-9.

[24]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operator Gleichungen und Operator Differential Gleichungen,", Akademie-Verlag, (1974).

[25]

H. Gajewski and K. Zacharias, Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum,, Math. Nachr., 57 (1973), 127. doi: 10.1002/mana.19730570107.

[26]

N. Ghoussoub, A variational theory for monotone vector fields,, J. Fixed Point Theory Appl., 4 (2008), 107. doi: 10.1007/s11784-008-0083-4.

[27]

N. Ghoussoub, "Selfdual Partial Differential Systems and Their Variational Principles,", Springer, (2009).

[28]

N. Ghoussoub and L. Tzou, A variational principle for gradient flows,, Math. Ann., 330 (2004), 519. doi: 10.1007/s00208-004-0558-6.

[29]

W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Springer, (1999).

[30]

A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems,", North-Holland, (1979).

[31]

H. Jian, On the homogenization of degenerate parabolic equations,, Acta Math. Appl. Sinica, 16 (2000), 100. doi: 10.1007/BF02670970.

[32]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,", Springer, (1989).

[33]

P. Krejčí, "Convexity, Hysteresis and Dissipation in Hyperbolic Equations,", Gakkotosho, (1997).

[34]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969).

[35]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vols. I, (1972).

[36]

J. E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions,, Set-Valued Anal., 13 (2005), 21. doi: 10.1007/s11228-004-4170-4.

[37]

J. E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product,, Proc. Amer. Math. Soc., 136 (2008), 873. doi: 10.1090/S0002-9939-07-09176-9.

[38]

J. E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators,, J. Nonlinear Convex Anal., 2 (2001), 243.

[39]

I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications,", Elsevier, (2003).

[40]

A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461.

[41]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl., 11 (2004), 151.

[42]

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. doi: 10.1007/s002050200194.

[43]

A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation,, Electron. J. Differential Equations, 1 (2001).

[44]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs,, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976).

[45]

J. P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules,, C. R. Math. Acad. Sci. Paris, 338 (2004), 853.

[46]

J. P. Penot, The relevance of convex analysis for the study of monotonicity,, Nonlinear Anal., 58 (2004), 855. doi: 10.1016/j.na.2004.05.018.

[47]

R. T. Rockafellar, A general correspondence between dual minimax problems and convex programs,, Pacific J. Math., 25 (1968), 597.

[48]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1969).

[49]

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., 7 (2008), 97.

[50]

T. Roubíček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (2005).

[51]

G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations,, Discrete Contin. Dyn. Syst., 18 (2007), 15. doi: 10.3934/dcds.2007.18.15.

[52]

T. Senba, On some nonlinear evolution equations,, Funkcial. Ekvac., 29 (1986), 243.

[53]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360.

[54]

U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations,, S. I. A. M. J. Control Optim., 8 (2008), 1615. doi: 10.1137/070684574.

[55]

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

[56]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994).

[57]

A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996).

[58]

A. Visintin, Quasilinear hyperbolic equations with hysteresis,, Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451.

[59]

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

[60]

A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators,, Adv. Math. Sci. Appl., 18 (2008), 633.

[61]

A. Visintin, Scale-transformations of maximal monotone relations in view of homogenization,, Boll. Un. Mat. Ital., III (2010), 591.

[62]

A. Visintin, Structural stability of doubly-nonlinear flows,, Boll. Un. Mat. Ital., IV (2011), 363.

[63]

A. Visintin, Variational formulation and structural stability of monotone equations,, Calc. Var. Partial Differential Equations (in press)., ().

show all references

References:
[1]

S. Aizicovici and Q. Yan, Convergence theorems for abstract doubly nonlinear differential equations,, Panamer. Math. J, 7 (1997), 1.

[2]

H. Attouch, "Variational Convergence for Functions and Operators,", Pitman, (1984).

[3]

G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations,, Differential Integral Equations, 6 (1993), 1161.

[4]

V. Barbu, Existence theorems for a class of two point boundary problems,, J. Differential Equations, 17 (1975), 236.

[5]

V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces,", Springer, (2010).

[6]

V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", Editura Academiei, (1978).

[7]

A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,", Oxford University Press, (1998).

[8]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (1973).

[9]

H. Brezis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983).

[10]

H. Brezis and I. Ekeland, Un principe variationnel associé àcertaines équations parabo-liques.I. Le cas indépendant du temps,, II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971.

[11]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). doi: 10.1007/978-1-4612-4048-8.

[12]

M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, J. Convex Anal., 15 (2008), 87.

[13]

R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements,, Set-Valued Analysis, 10 (2002), 297. doi: 10.1023/A:1020639314056.

[14]

R. S. Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product,, Proc. Amer. Math. Soc., 131 (2003), 2379. doi: 10.1090/S0002-9939-03-07053-9.

[15]

P. Colli, On some doubly nonlinear evolution equations in Banach spaces,, Japan J. Indust. Appl. Math., 9 (1992), 181. doi: 10.1007/BF03167565.

[16]

P. Colli and A. Visintin, On a class of doubly nonlinear evolution problems,, Communications in P. D. E. s, 15 (1990), 737.

[17]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993).

[18]

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. doi: 10.1007/s002050100187.

[19]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842.

[20]

I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles,", Dunod Gau-thier-Villars, (1974).

[21]

W. Fenchel, "Convex Cones, Sets, and Functions,", Princeton Univ., (1953).

[22]

S. Fitzpatrick, Representing monotone operators by convex functions,, Workshop/Minicon-ference on Functional Analysis and Optimization (Canberra, (1988), 59.

[23]

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. doi: 10.1016/S0022-5096(98)00034-9.

[24]

H. Gajewski, K. Gröger and K. Zacharias, "Nichtlineare Operator Gleichungen und Operator Differential Gleichungen,", Akademie-Verlag, (1974).

[25]

H. Gajewski and K. Zacharias, Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum,, Math. Nachr., 57 (1973), 127. doi: 10.1002/mana.19730570107.

[26]

N. Ghoussoub, A variational theory for monotone vector fields,, J. Fixed Point Theory Appl., 4 (2008), 107. doi: 10.1007/s11784-008-0083-4.

[27]

N. Ghoussoub, "Selfdual Partial Differential Systems and Their Variational Principles,", Springer, (2009).

[28]

N. Ghoussoub and L. Tzou, A variational principle for gradient flows,, Math. Ann., 330 (2004), 519. doi: 10.1007/s00208-004-0558-6.

[29]

W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Springer, (1999).

[30]

A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems,", North-Holland, (1979).

[31]

H. Jian, On the homogenization of degenerate parabolic equations,, Acta Math. Appl. Sinica, 16 (2000), 100. doi: 10.1007/BF02670970.

[32]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,", Springer, (1989).

[33]

P. Krejčí, "Convexity, Hysteresis and Dissipation in Hyperbolic Equations,", Gakkotosho, (1997).

[34]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod, (1969).

[35]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Vols. I, (1972).

[36]

J. E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions,, Set-Valued Anal., 13 (2005), 21. doi: 10.1007/s11228-004-4170-4.

[37]

J. E. Martinez-Legaz and B. F. Svaiter, Minimal convex functions bounded below by the duality product,, Proc. Amer. Math. Soc., 136 (2008), 873. doi: 10.1090/S0002-9939-07-09176-9.

[38]

J. E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators,, J. Nonlinear Convex Anal., 2 (2001), 243.

[39]

I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications,", Elsevier, (2003).

[40]

A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461.

[41]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl., 11 (2004), 151.

[42]

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. doi: 10.1007/s002050200194.

[43]

A. K. Nandakumaran and M. Rajesh, Homogenization of a nonlinear degenerate parabolic differential equation,, Electron. J. Differential Equations, 1 (2001).

[44]

B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs,, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976).

[45]

J. P. Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules,, C. R. Math. Acad. Sci. Paris, 338 (2004), 853.

[46]

J. P. Penot, The relevance of convex analysis for the study of monotonicity,, Nonlinear Anal., 58 (2004), 855. doi: 10.1016/j.na.2004.05.018.

[47]

R. T. Rockafellar, A general correspondence between dual minimax problems and convex programs,, Pacific J. Math., 25 (1968), 597.

[48]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1969).

[49]

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., 7 (2008), 97.

[50]

T. Roubíček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (2005).

[51]

G. Schimperna, A. Segatti and U. Stefanelli, Well-posedness and long-time behavior for a class of doubly nonlinear equations,, Discrete Contin. Dyn. Syst., 18 (2007), 15. doi: 10.3934/dcds.2007.18.15.

[52]

T. Senba, On some nonlinear evolution equations,, Funkcial. Ekvac., 29 (1986), 243.

[53]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360.

[54]

U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations,, S. I. A. M. J. Control Optim., 8 (2008), 1615. doi: 10.1137/070684574.

[55]

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

[56]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994).

[57]

A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996).

[58]

A. Visintin, Quasilinear hyperbolic equations with hysteresis,, Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451.

[59]

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

[60]

A. Visintin, Extension of the Brezis-Ekeland-Nayroles principle to monotone operators,, Adv. Math. Sci. Appl., 18 (2008), 633.

[61]

A. Visintin, Scale-transformations of maximal monotone relations in view of homogenization,, Boll. Un. Mat. Ital., III (2010), 591.

[62]

A. Visintin, Structural stability of doubly-nonlinear flows,, Boll. Un. Mat. Ital., IV (2011), 363.

[63]

A. Visintin, Variational formulation and structural stability of monotone equations,, Calc. Var. Partial Differential Equations (in press)., ().

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