# American Institute of Mathematical Sciences

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 and 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), 1-17. [2] H. Attouch, "Variational Convergence for Functions and Operators," Pitman, Boston, 1984. [3] G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-117. [4] V. Barbu, Existence theorems for a class of two point boundary problems, J. Differential Equations, 17 (1975), 236-257. [5] V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces," Springer, Berlin, 2010. [6] V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces," Editura Academiei, Bucuresti, 1978. [7] A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals," Oxford University Press, Oxford, 1998. [8] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973. [9] H. Brezis, "Analyse Fonctionelle. Théorie et Applications," Masson, Paris, 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), 971-974, and 1197-1198. [11] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, Berlin, 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), 87-104. [13] R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements, Set-Valued Analysis, 10 (2002), 297-316. 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), 2379-2383. 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), 181-203. 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), 737-756. [17] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Boston, 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), 101-135. 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), 842-850. [20] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles," Dunod Gau-thier-Villars, Paris, 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), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. [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), 1319-1342. 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, Berlin, 1974. [25] H. Gajewski and K. Zacharias, Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum, Math. Nachr., 57 (1973), 127-140. doi: 10.1002/mana.19730570107. [26] N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl., 4 (2008), 107-135. 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), 519-549. doi: 10.1007/s00208-004-0558-6. [29] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Springer, New York, 1999. [30] A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems," North-Holland, Amsterdam, 1979. [31] H. Jian, On the homogenization of degenerate parabolic equations, Acta Math. Appl. Sinica, 16 (2000), 100-110. doi: 10.1007/BF02670970. [32] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis," Springer, Berlin 1989. (Russian ed. Nauka, Moscow 1983). [33] P. Krejčí, "Convexity, Hysteresis and Dissipation in Hyperbolic Equations," Gakkotosho, Tokyo, 1997. [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, Paris, 1969. [35] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," Vols. I,II. Springer, Berlin 1972. (French edition: Dunod, Paris 1968). [36] J. E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), 21-46. 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), 873-878. 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), 243-247. [39] I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications," Elsevier, Amsterdam, 2003. [40] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [41] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl., 11 (2004), 151-189. [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), 137-177. 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), 19 pp. [44] B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038. [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, Ser. I, 338 (2004), 853-858. [46] J. P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), 855-871. 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), 597-611. [48] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 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), 97-169. [50] T. Roubíček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 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), 15-38. doi: 10.3934/dcds.2007.18.15. [52] T. Senba, On some nonlinear evolution equations, Funkcial. Ekvac., 29 (1986), 243-257. [53] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [54] U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, S. I. A. M. J. Control Optim., 8 (2008), 1615-1642. doi: 10.1137/070684574. [55] A. Visintin, A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5-18. [56] A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994. [57] A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996. [58] A. Visintin, Quasilinear hyperbolic equations with hysteresis, Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451-476. [59] A. Visintin, Maxwell's equations with vector hysteresis, Arch. Rat. Mech. Anal., 175 (2005), 1-38. 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), 633-650. [61] A. Visintin, Scale-transformations of maximal monotone relations in view of homogenization, Boll. Un. Mat. Ital., III (2010), 591-601 [62] A. Visintin, Structural stability of doubly-nonlinear flows, Boll. Un. Mat. Ital., IV (2011), 363-391. [63] A. Visintin, Variational formulation and structural stability of monotone equations,, Calc. Var. Partial Differential Equations (in press)., ().

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##### References:
 [1] S. Aizicovici and Q. Yan, Convergence theorems for abstract doubly nonlinear differential equations, Panamer. Math. J, 7 (1997), 1-17. [2] H. Attouch, "Variational Convergence for Functions and Operators," Pitman, Boston, 1984. [3] G. Auchmuty, Saddle-points and existence-uniqueness for evolution equations, Differential Integral Equations, 6 (1993), 1161-117. [4] V. Barbu, Existence theorems for a class of two point boundary problems, J. Differential Equations, 17 (1975), 236-257. [5] V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces," Springer, Berlin, 2010. [6] V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces," Editura Academiei, Bucuresti, 1978. [7] A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals," Oxford University Press, Oxford, 1998. [8] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973. [9] H. Brezis, "Analyse Fonctionelle. Théorie et Applications," Masson, Paris, 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), 971-974, and 1197-1198. [11] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, Berlin, 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), 87-104. [13] R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions, and a special family of enlargements, Set-Valued Analysis, 10 (2002), 297-316. 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), 2379-2383. 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), 181-203. 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), 737-756. [17] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Boston, 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), 101-135. 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), 842-850. [20] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles," Dunod Gau-thier-Villars, Paris, 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), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. [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), 1319-1342. 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, Berlin, 1974. [25] H. Gajewski and K. Zacharias, Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum, Math. Nachr., 57 (1973), 127-140. doi: 10.1002/mana.19730570107. [26] N. Ghoussoub, A variational theory for monotone vector fields, J. Fixed Point Theory Appl., 4 (2008), 107-135. 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), 519-549. doi: 10.1007/s00208-004-0558-6. [29] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Springer, New York, 1999. [30] A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems," North-Holland, Amsterdam, 1979. [31] H. Jian, On the homogenization of degenerate parabolic equations, Acta Math. Appl. Sinica, 16 (2000), 100-110. doi: 10.1007/BF02670970. [32] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis," Springer, Berlin 1989. (Russian ed. Nauka, Moscow 1983). [33] P. Krejčí, "Convexity, Hysteresis and Dissipation in Hyperbolic Equations," Gakkotosho, Tokyo, 1997. [34] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod, Paris, 1969. [35] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," Vols. I,II. Springer, Berlin 1972. (French edition: Dunod, Paris 1968). [36] J. E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13 (2005), 21-46. 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), 873-878. 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), 243-247. [39] I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications," Elsevier, Amsterdam, 2003. [40] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [41] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl., 11 (2004), 151-189. [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), 137-177. 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), 19 pp. [44] B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038. [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, Ser. I, 338 (2004), 853-858. [46] J. P. Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., 58 (2004), 855-871. 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), 597-611. [48] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 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), 97-169. [50] T. Roubíček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 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), 15-38. doi: 10.3934/dcds.2007.18.15. [52] T. Senba, On some nonlinear evolution equations, Funkcial. Ekvac., 29 (1986), 243-257. [53] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [54] U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, S. I. A. M. J. Control Optim., 8 (2008), 1615-1642. doi: 10.1137/070684574. [55] A. Visintin, A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5-18. [56] A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994. [57] A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996. [58] A. Visintin, Quasilinear hyperbolic equations with hysteresis, Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451-476. [59] A. Visintin, Maxwell's equations with vector hysteresis, Arch. Rat. Mech. Anal., 175 (2005), 1-38. 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), 633-650. [61] A. Visintin, Scale-transformations of maximal monotone relations in view of homogenization, Boll. Un. Mat. Ital., III (2010), 591-601 [62] A. Visintin, Structural stability of doubly-nonlinear flows, Boll. Un. Mat. Ital., IV (2011), 363-391. [63] A. Visintin, Variational formulation and structural stability of monotone equations,, Calc. Var. Partial Differential Equations (in press)., ().
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