Weighted energy-dissipation functionals for gradient flows

Alexander Mielke; Ulisse Stefanelli

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 52-85
  • ISSN: 1292-8119

Abstract

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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.].

How to cite

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Mielke, Alexander, and Stefanelli, Ulisse. "Weighted energy-dissipation functionals for gradient flows." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 52-85. <http://eudml.org/doc/276333>.

@article{Mielke2011,
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.]. },
author = {Mielke, Alexander, Stefanelli, Ulisse},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Variational principle; gradient flow; convergence; abstract evolution; microstructure evolution},
language = {eng},
month = {2},
number = {1},
pages = {52-85},
publisher = {EDP Sciences},
title = {Weighted energy-dissipation functionals for gradient flows},
url = {http://eudml.org/doc/276333},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Mielke, Alexander
AU - Stefanelli, Ulisse
TI - Weighted energy-dissipation functionals for gradient flows
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 52
EP - 85
AB - 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.].
LA - eng
KW - Variational principle; gradient flow; convergence; abstract evolution; microstructure evolution
UR - http://eudml.org/doc/276333
ER -

References

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  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.  
  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 Wissenschaften223. 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.  
  6. D. Brézis, Classes d'interpolation associées à un opérateur monotone. C. R. Acad. Sci. Paris Sér. A-B276 (1973) A1553–A1556.  
  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. Studies5. 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-B282 (1976) A971–A974.  
  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-B282 (1976) A1197–A1198.  
  12. F.H. Clarke, Optimization and nonsmooth analysis, Classics in Applied Mathematics5. Second edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (1990).  
  13. S. Conti and M. Ortiz, Minimum principles for the trajectories of systems governed by rate problems. J. Mech. Phys. Solids56 (2008) 1885–1904.  
  14. M.G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets. J. Funct. Anal.3 (1969) 376–418.  
  15. E. De Giorgi, Conjectures concerning some evolution problems. Duke Math. J.81 (1996) 255–268. A celebration of John F. Nash, Jr.  
  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.  
  18. M.E. Gurtin, Variational principles for linear elastodynamics. Arch. Ration. Mech. Anal.16 (1964) 34–50.  
  19. M.E. Gurtin, Variational principles for linear initial value problems. Quart. Appl. Math.22 (1964) 252–256.  
  20. I. Hlaváček, Variational principles for parabolic equations. Appl. Math.14 (1969) 278–297.  
  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.  
  23. Y. Kōmura, Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan19 (1967) 493–507.  
  24. J.-L. Lions and E. Magenes, Non-homogeneus boundary value problems and applications1. 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.  
  26. A. Mielke and M. Ortiz, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM: COCV14 (2008) 494–516.  
  27. A. Mielke and U. Stefanelli, A discrete variational principle for rate-independent evolution. Adv. Calc. Var.1 (2008) 399–431.  
  28. B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B282 (1976) A1035–A1038.  
  29. B. Nayroles, Un théorème de minimum pour certains systèmes dissipatifs. Variante hilbertienne. Travaux Sém. Anal. Convexe6 (1976) 22.  
  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.  
  31. M. Ortiz, E.A. Repetto and H. Si, A continuum model of kinetic roughening and coarsening in thin films. J. Mech. Phys. Solids47 (1999) 697–730.  
  32. F. Otto, The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations26 (2001) 101–174.  
  33. R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM: COCV12 (2006) 564–614.  
  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.  
  36. G. Savaré, Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl.6 (1996) 377–418.  
  37. J. Simon, Compact sets in the space Lp(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.  
  39. U. Stefanelli, A variational principle for hardening elasto-plasticity. SIAM J. Math. Anal.40 (2008) 623–652.  
  40. U. Stefanelli, The discrete Brezis-Ekeland principle. J. Convex Anal.16 (2009) 71–87.  
  41. L. Tartar, Théorème d'interpolation non linéaire et applications. C. R. Acad. Sci. Paris Sér. A-B270 (1970) A1729–A1731.  
  42. L. Tartar, Interpolation non linéaire et régularité. J. Funct. Anal.9 (1972) 469–489.  
  43. H. Triebel, Interpolation theory, function spaces, differential operators. Second edition, Johann Ambrosius Barth, Heidelberg, Germany (1995).  
  44. A. Visintin, A new approach to evolution. C. R. Acad. Sci. Paris Sér. I Math.332 (2001) 233–238.  
  45. A. Visintin, An extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl.18 (2008) 633–650.  

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