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Weighted energy-dissipation functionals for gradient flows

Published online by Cambridge University Press:  30 October 2009

Alexander Mielke  and
Ulisse Stefanelli
Alexander Mielke
Affiliation:
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany. [email protected] Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany.
Ulisse Stefanelli
Affiliation:
IMATI – CNR, v. Ferrata 1, 27100 Pavia, Italy. [email protected]
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Abstract

We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke and Ortiz [ESAIM: COCV14 (2008) 494–516]. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from [S. Conti and M. Ortiz, J. Mech. Phys. Solids56 (2008) 1885–1904.].

Keywords

Variational principlegradient flowconvergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 52 - 85
Copyright
© EDP Sciences, SMAI, 2009

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References

L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, ETH Zürich. Birkhäuser Verlag, Basel, Switzerland (2005).
Baiocchi, C. and Savaré, G., Singular perturbation and interpolation. Math. Models Methods Appl. Sci. 4 (1994) 557–570. CrossRef
V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publishing, Leyden, The Netherlands (1976).
J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften 223. Springer-Verlag, Berlin, Germany (1976).
Biot, M.A., Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys. Rev. (2) 97 (1955) 1463–1469. CrossRef
Brézis, D., Classes d'interpolation associées à un opérateur monotone. C. R. Acad. Sci. Paris Sér. A-B 276 (1973) A1553–A1556.
H. Brezis, Monotonicity methods in Hilbert spaces and some application to nonlinear partial differential equations, in Contrib. to nonlin. functional analysis, Proc. Sympos. Univ. Wisconsin, Madison, Academic Press, New York, USA (1971) 101–156.
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland Math. Studies 5. Amsterdam, North-Holland (1973).
H. Brezis, Interpolation classes for monotone operators, in Partial differential equations and related topics (Program, Tulane Univ., New Orleans, 1974), Lecture Notes in Math. 446, Springer, Berlin, Germany (1975) 65–74.
Brezis, H. and Ekeland, I., Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 282 (1976) A971–A974.
Brezis, H. and Ekeland, I., Un principe variationnel associé à certaines équations paraboliques. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 282 (1976) A1197–A1198.
F.H. Clarke, Optimization and nonsmooth analysis, Classics in Applied Mathematics 5. Second edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1990).
Conti, S. and Ortiz, M., Minimum principles for the trajectories of systems governed by rate problems. J. Mech. Phys. Solids 56 (2008) 1885–1904. CrossRef
Crandall, M.G. and Pazy, A., Semi-groups of nonlinear contractions and dissipative sets. J. Funct. Anal. 3 (1969) 376–418. CrossRef
De Giorgi, E., Conjectures concerning some evolution problems. Duke Math. J. 81 (1996) 255–268. A celebration of John F. Nash, Jr. CrossRef
N. Ghoussoub, Selfdual partial differential systems and their variational principles, Springer Monographs in Mathematics. Springer, New York, USA (2009).
Gurtin, M.E., Variational principles in the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 13 (1963) 179–191. CrossRef
Gurtin, M.E., Variational principles for linear elastodynamics. Arch. Ration. Mech. Anal. 16 (1964) 34–50. CrossRef
Gurtin, M.E., Variational principles for linear initial value problems. Quart. Appl. Math. 22 (1964) 252–256. CrossRef
Hlaváček, I., Variational principles for parabolic equations. Appl. Math. 14 (1969) 278–297.
T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108. American Mathematical Society, USA (1994).
Kohn, R.V. and Müller, S., Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47 (1994) 405–435. CrossRef
Kōmura, Y., Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan 19 (1967) 493–507. CrossRef
J.-L. Lions and E. Magenes, Non-homogeneus boundary value problems and applications 1. Springer-Verlag, New York-Heidelberg (1972).
Marino, A., Saccon, C. and Tosques, M., Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989) 281–330.
Mielke, A. and Ortiz, M., A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM: COCV 14 (2008) 494–516. CrossRef
Mielke, A. and Stefanelli, U., A discrete variational principle for rate-independent evolution. Adv. Calc. Var. 1 (2008) 399–431. CrossRef
Nayroles, B., Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B 282 (1976) A1035–A1038.
Nayroles, B., Un théorème de minimum pour certains systèmes dissipatifs. Variante hilbertienne. Travaux Sém. Anal. Convexe 6 (1976) 22.
Nochetto, R., Savaré, G. and Verdi, C., A posteriori error estimates for variable time-step discretization of nonlinear evolution equations. Comm. Pure Appl. Math. 53 (2000) 525–589. 3.0.CO;2-M>CrossRef
Ortiz, M., Repetto, E.A. and Si, H., A continuum model of kinetic roughening and coarsening in thin films. J. Mech. Phys. Solids 47 (1999) 697–730. CrossRef
Otto, F., The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101–174. CrossRef
Rossi, R. and Savaré, G., Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: COCV 12 (2006) 564–614. CrossRef
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. (5) VII (2008) 97–169.
Rossi, R., Segatti, A. and Stefanelli, U., Attractors for gradient flows of non convex functionals and applications. Arch. Ration. Anal. Mech. 187 (2008) 91–135. CrossRef
Savaré, G., Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6 (1996) 377–418.
J. Simon, Compact sets in the space L p (0, T; B). Ann. Mat. Pura Appl. (4) 146 (1987) 65–96.
Stefanelli, U., The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Contr. Opt. 47 (2008) 1615–1642. CrossRef
Stefanelli, U., A variational principle for hardening elasto-plasticity. SIAM J. Math. Anal. 40 (2008) 623–652. CrossRef
Stefanelli, U., The discrete Brezis-Ekeland principle. J. Convex Anal. 16 (2009) 71–87.
Tartar, L., Théorème d'interpolation non linéaire et applications. C. R. Acad. Sci. Paris Sér. A-B 270 (1970) A1729–A1731.
Tartar, L., Interpolation non linéaire et régularité. J. Funct. Anal. 9 (1972) 469–489. CrossRef
H. Triebel, Interpolation theory, function spaces, differential operators. Second edition, Johann Ambrosius Barth, Heidelberg, Germany (1995).
Visintin, A., A new approach to evolution. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 233–238. CrossRef
Visintin, A., An extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl. 18 (2008) 633–650.