ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2011 - Volume 17 - Issue01

Research Article

Weighted energy-dissipation functionals for gradient flows

Mielke, Alexandera1a2 and Stefanelli, Ulissea3

a1 Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany. mielke@wias-berlin.de

a2 Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany.

a3 IMATI – CNR, v. Ferrata 1, 27100 Pavia, Italy. ulisse.stefanelli@imati.cnr.it

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: COCV 14 (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. Solids 56 (2008) 1885–1904.].

(Received January 30 2009)

(Online publication October 30 2009)

Key Words:

  • Variational principle;
  • gradient flow;
  • convergence

Mathematics Subject Classification:

  • 35K55
  • [1] 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).
  • [2] C. Baiocchi and G. Savaré , Singular perturbation and interpolation. Math. Models Methods Appl. Sci. 4 (1994) 557–570. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [3] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publishing, Leyden, The Netherlands (1976).
  • [4] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften 223. Springer-Verlag, Berlin, Germany (1976).
  • [5] M.A. Biot , Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys. Rev. (2) 97 (1955) 1463–1469. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [6] D. Brézis , Classes d'interpolation associées à un opérateur monotone. C. R. Acad. Sci. Paris Sér. A-B 276 (1973) A1553–A1556. [OpenURL Query Data]  [Google Scholar]
  • [7] 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.
  • [8] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland Math. Studies 5. Amsterdam, North-Holland (1973).
  • [9] 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.
  • [10] H. Brezis and I. Ekeland , 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. [OpenURL Query Data]  [Google Scholar]
  • [11] H. Brezis and I. Ekeland , 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. [OpenURL Query Data]  [Google Scholar]
  • [12] F.H. Clarke, Optimization and nonsmooth analysis, Classics in Applied Mathematics 5. Second edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1990). [Google Scholar]
  • [13] S. Conti and M. Ortiz , Minimum principles for the trajectories of systems governed by rate problems. J. Mech. Phys. Solids 56 (2008) 1885–1904. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [14] M.G. Crandall and A. Pazy , Semi-groups of nonlinear contractions and dissipative sets. J. Funct. Anal. 3 (1969) 376–418. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [15] E. De Giorgi , Conjectures concerning some evolution problems. Duke Math. J. 81 (1996) 255–268. A celebration of John F. Nash, Jr. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [16] N. Ghoussoub, Selfdual partial differential systems and their variational principles, Springer Monographs in Mathematics. Springer, New York, USA (2009).
  • [17] M.E. Gurtin , Variational principles in the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 13 (1963) 179–191. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [18] M.E. Gurtin , Variational principles for linear elastodynamics. Arch. Ration. Mech. Anal. 16 (1964) 34–50. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [19] M.E. Gurtin , Variational principles for linear initial value problems. Quart. Appl. Math. 22 (1964) 252–256. [OpenURL Query Data]  [Google Scholar]
  • [20] I. Hlaváček , Variational principles for parabolic equations. Appl. Math. 14 (1969) 278–297. [OpenURL Query Data]  [Google Scholar]
  • [21] T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108. American Mathematical Society, USA (1994).
  • [22] R.V. Kohn and S. Müller , Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47 (1994) 405–435. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [23] Y. Kōmura , Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan 19 (1967) 493–507. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [24] J.-L. Lions and E. Magenes, Non-homogeneus boundary value problems and applications 1. Springer-Verlag, New York-Heidelberg (1972).
  • [25] A. Marino , C. Saccon and M. Tosques , Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989) 281–330. [OpenURL Query Data]  [Google Scholar]
  • [26] A. Mielke and M. Ortiz , A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM: COCV 14 (2008) 494–516. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [27] A. Mielke and U. Stefanelli , A discrete variational principle for rate-independent evolution. Adv. Calc. Var. 1 (2008) 399–431. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [28] B. Nayroles , Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B 282 (1976) A1035–A1038. [OpenURL Query Data]  [Google Scholar]
  • [29] B. Nayroles , Un théorème de minimum pour certains systèmes dissipatifs. Variante hilbertienne. Travaux Sém. Anal. Convexe 6 (1976) 22. [OpenURL Query Data]  [Google Scholar]
  • [30] R. Nochetto , G. Savaré and C. Verdi , A posteriori error estimates for variable time-step discretization of nonlinear evolution equations. Comm. Pure Appl. Math. 53 (2000) 525–589. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [31] M. Ortiz , E.A. Repetto and H. Si , A continuum model of kinetic roughening and coarsening in thin films. J. Mech. Phys. Solids 47 (1999) 697–730. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [32] F. Otto , The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101–174. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [33] R. Rossi and G. Savaré , Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: COCV 12 (2006) 564–614. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [34] 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.
  • [35] R. Rossi , A. Segatti and U. Stefanelli , Attractors for gradient flows of non convex functionals and applications. Arch. Ration. Anal. Mech. 187 (2008) 91–135. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [36] G. Savaré , Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6 (1996) 377–418. [OpenURL Query Data]  [Google Scholar]
  • [37] J. Simon, Compact sets in the space L p (0, T; B). Ann. Mat. Pura Appl. (4) 146 (1987) 65–96.
  • [38] U. Stefanelli , The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Contr. Opt. 47 (2008) 1615–1642. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [39] U. Stefanelli , A variational principle for hardening elasto-plasticity. SIAM J. Math. Anal. 40 (2008) 623–652. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [40] U. Stefanelli , The discrete Brezis-Ekeland principle. J. Convex Anal. 16 (2009) 71–87. [OpenURL Query Data]  [Google Scholar]
  • [41] L. Tartar , Théorème d'interpolation non linéaire et applications. C. R. Acad. Sci. Paris Sér. A-B 270 (1970) A1729–A1731. [OpenURL Query Data]  [Google Scholar]
  • [42] L. Tartar , Interpolation non linéaire et régularité. J. Funct. Anal. 9 (1972) 469–489. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [43] H. Triebel, Interpolation theory, function spaces, differential operators. Second edition, Johann Ambrosius Barth, Heidelberg, Germany (1995). [Google Scholar]
  • [44] A. Visintin , A new approach to evolution. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 233–238. [OpenURL Query Data]  [Google Scholar]
  • [45] A. Visintin , An extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl. 18 (2008) 633–650. [OpenURL Query Data]  [Google Scholar]