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
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show all references

References:
[1]

Panamer. Math. J, 7 (1997), 1-17.  Google Scholar

[2]

Pitman, Boston, 1984.  Google Scholar

[3]

Differential Integral Equations, 6 (1993), 1161-117.  Google Scholar

[4]

J. Differential Equations, 17 (1975), 236-257.  Google Scholar

[5]

Springer, Berlin, 2010.  Google Scholar

[6]

Editura Academiei, Bucuresti, 1978.  Google Scholar

[7]

Oxford University Press, Oxford, 1998.  Google Scholar

[8]

North-Holland, Amsterdam, 1973.  Google Scholar

[9]

Masson, Paris, 1983.  Google Scholar

[10]

II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974, and 1197-1198.  Google Scholar

[11]

Springer, Berlin, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[12]

J. Convex Anal., 15 (2008), 87-104.  Google Scholar

[13]

Set-Valued Analysis, 10 (2002), 297-316. doi: 10.1023/A:1020639314056.  Google Scholar

[14]

Proc. Amer. Math. Soc., 131 (2003), 2379-2383. doi: 10.1090/S0002-9939-03-07053-9.  Google Scholar

[15]

Japan J. Indust. Appl. Math., 9 (1992), 181-203. doi: 10.1007/BF03167565.  Google Scholar

[16]

Communications in P. D. E. s, 15 (1990), 737-756.  Google Scholar

[17]

Birkhäuser, Boston, 1993.  Google Scholar

[18]

Arch. Rat. Mech. Anal., 162 (2002), 101-135. doi: 10.1007/s002050100187.  Google Scholar

[19]

Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850.  Google Scholar

[20]

Dunod Gau-thier-Villars, Paris, 1974.  Google Scholar

[21]

Princeton Univ., 1953. Google Scholar

[22]

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.  Google Scholar

[23]

J. Mech. Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[24]

Akademie-Verlag, Berlin, 1974.  Google Scholar

[25]

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[26]

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[27]

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[28]

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[29]

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[30]

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[31]

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Springer, Berlin 1989. (Russian ed. Nauka, Moscow 1983).  Google Scholar

[33]

Gakkotosho, Tokyo, 1997. Google Scholar

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Dunod, Paris, 1969.  Google Scholar

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Vols. I,II. Springer, Berlin 1972. (French edition: Dunod, Paris 1968).  Google Scholar

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[38]

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Elsevier, Amsterdam, 2003. Google Scholar

[40]

in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.  Google Scholar

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Nonl. Diff. Eqns. Appl., 11 (2004), 151-189.  Google Scholar

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[43]

Electron. J. Differential Equations, 1 (2001), 19 pp.  Google Scholar

[44]

C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038.  Google Scholar

[45]

C. R. Math. Acad. Sci. Paris, Ser. I, 338 (2004), 853-858.  Google Scholar

[46]

Nonlinear Anal., 58 (2004), 855-871. doi: 10.1016/j.na.2004.05.018.  Google Scholar

[47]

Pacific J. Math., 25 (1968), 597-611.  Google Scholar

[48]

Princeton University Press, Princeton, 1969.  Google Scholar

[49]

Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 97-169.  Google Scholar

[50]

Birkhäuser, Basel, 2005.  Google Scholar

[51]

Discrete Contin. Dyn. Syst., 18 (2007), 15-38. doi: 10.3934/dcds.2007.18.15.  Google Scholar

[52]

Funkcial. Ekvac., 29 (1986), 243-257.  Google Scholar

[53]

Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[54]

S. I. A. M. J. Control Optim., 8 (2008), 1615-1642. doi: 10.1137/070684574.  Google Scholar

[55]

Control and Cybernetics, 11 (1982), 5-18.  Google Scholar

[56]

Springer, Berlin, 1994.  Google Scholar

[57]

Birkhäuser, Boston, 1996.  Google Scholar

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Ann. Inst. H. Poincaré. Analyse non lineaire, 19 (2002), 451-476.  Google Scholar

[59]

Arch. Rat. Mech. Anal., 175 (2005), 1-38. doi: 10.1007/s00205-004-0333-6.  Google Scholar

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Boll. Un. Mat. Ital., III (2010), 591-601  Google Scholar

[62]

Boll. Un. Mat. Ital., IV (2011), 363-391.  Google Scholar

[63]

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

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