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Erschienen in:
On the Dynamics of Large Particle Systems in the Mean Field Limit Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

3. On Evolutionary \(\varGamma \)-Convergence for Gradient Systems

verfasst von : Alexander Mielke

Erschienen in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

Verlag: Springer International Publishing

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Abstract

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional \({\mathscr {E}}_\varepsilon \) and the dissipation potential \({\mathscr {R}}_\varepsilon \) or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions \(u_\varepsilon \). We investigate the question under which conditions the limits u of (subsequences of) the solutions \(u_\varepsilon \) are solutions of the gradient system generated by the \(\varGamma \)-limits \({\mathscr {E}}_0\) and \({\mathscr {R}}_0\). Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where \({\mathscr {R}}_\varepsilon \) is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.

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Vorheriges Kapitel Continuum Limits of Discrete Models via -Convergence
Nächstes Kapitel Homoclinic Points of Principal Algebraic Actions
Literatur
1.
Alber, H.-D.: Materials with memory. Lecture Notes in Mathematics, vol. 1682. Springer-Verlag, Berlin (1998)
2.
Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005)
3.
Arnrich, S., Mielke, A., Peletier, M.A., Savaré, G., Veneroni, M.: Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction. Calc. Var. Part. Diff. Eqns. 44, 419–454 (2012)MATHCrossRef
4.
Attouch, H.: Variational Convergence of Functions and Operators. Pitman Advanced Publishing Program, Pitman (1984)
5.
Bellettini, G., Bertini, L., Mariani, M., Novaga, M.: Convergence of the one-dimensional Cahn-Hilliard equation. SIAM J. Math. Anal. 44(5), 3458–3480 (2012)MATHMathSciNetCrossRef
6.
Bénilan, P.: Solutions intégrales d’équations d’évolution dans un espace de Banach, C. R. Acad. Sci. Paris Sér. A-B 274, A47–A50 (1972)
7.
Biot, M.A.: Variational principles in irreversible thermodynamics with applications to viscoelasticity. Phys. Rev. 97(6), 1463–1469 (1955)MATHMathSciNetCrossRef
8.
Braides, A.: G-Convergence for Beginners. Oxford University Press (2002)
9.
Braides, A.: A handbook of g-convergence. In: Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 3 (2006)
10.
Braides, A.: Local minimization, variational evolution and gamma-convergence. Lecture Notes in Mathematics, vol. 2094. Springer (2013)
11.
Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973)MATH
12.
Brokate, M., Krejč’i, P., Schnabel, H.: On uniqueness in evolution quasivariational inequalities. J. Convex Anal. 11, 111–130 (2004)MATHMathSciNet
13.
Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176, 165–225 (2005)
14.
Dal Maso, G.: An Introduction to G-convergence. Birkhäuser Boston Inc., Boston (1993)
15.
Daneri, S., Savaré, G.: Lecture notes on gradient flows and optimal transport (2010)
16.
De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850 (1975)
17.
De Giorgi, E., Marino, A., Tosques, M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187 (1980)
18.
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North Holland (1976)
19.
Feireisl, E., Novotný, A. :Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser (2009)
20.
21.
Francfort, G.A., Larsen, C.J.: Existence and convergence for quasi-static evolution of brittle fracture. Commun. Pure Appl. Math. 56, 1495–1500 (2003)MathSciNetCrossRef
22.
Giacomini, A.: Ambrosio-Tortorelli approximation of quasistatic evolution of brittle fractures. Calc. Var. Part. Diff. Eqns. 22, 129–172 (2005)MATHMathSciNetCrossRef
23.
Giacomini, A., Ponsiglione, M.: AG-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications. Arch. Ration. Mech. Anal. 180(3), 399–447 (2006)MATHMathSciNetCrossRef
24.
Giacomini, A., Musesti, A.: Two-scale homogenization for a model in strain gradient plasticity. ESAIM Control Optim. Calc. Var. 17(4), 1035–1065 (2011). Published online. doi:10.​1051/​cocv/​2010036
25.
Giacomini, A., Ponsiglione, M.: Discontinuous finite element approximation of quasistatic crack growth in nonlinear elasticity. Math. Models Meth. Appl. Sci. (M3AS) 16, 77–118 (2006)
26.
Gigli, N., Maas, J.: Gromov-Hausdorff convergence of discrete transportation metrics. SIAM J. Math. Anal. 45(2), 879–899 (2013)MATHMathSciNetCrossRef
27.
Glitzky, A.: An electronic model for solar cells including active interfaces and energy resolved defect densities. SIAM J. Math. Anal. 44, 3874–3900 (2012)MATHMathSciNetCrossRef
28.
Glitzky, A., Mielke, A.: A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces. Z. Angew. Math. Phys. (ZAMP) 64, 29–52 (2013)MATHMathSciNetCrossRef
29.
30.
31.
James, R.D.: Hysteresis in Phase Transformations, Iciam 95 (hamburg, 1995), pp. 135–154 1996
32.
Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MATHMathSciNetCrossRef
33.
Krejč’i, P.: Evolution variational inequalities and multidimensional hysteresis operators. In: Nonlinear Differential Equations, pp. 47–110 (1999)
34.
Le, N.Q.: A gamma-convergence approach to the Cahn-Hilliard equation. Calc. Var. Part. Diff. Eqns. 32(4), 499–522 (2008)MATHCrossRef
35.
Liero, M.: Variational methods for evolution. Ph.D. Thesis (2012)
36.
Liero, M.: Passing from bulk to bulk-surface evolution in the Allen-Cahn equation. Nonl. Diff. Eqns. Appl. (NoDEA) 20(3), 919–942 (2013)
37.
Liero, M., Mielke, A.: An evolutionary elastoplastic plate model derived via G-convergence. Math. Models Meth. Appl. Sci. 21(9), 1961–1986 (2011)MATHMathSciNetCrossRef
38.
Liero, M., Mielke, A., Peletier, M.A., Renger, D.R.M.: On the microscopic origin of generalized gradient structures. In preparation (2015)
39.
Liero, M., Mielke, A., Savaré, G. :Optimal transport in competition with reaction-the Hellinger-Kantorovich distance. In preparation (2014)
40.
41.
Mielke, A.: Evolution in rate-independent systems (Ch. 6). In: Handbook of Differential Equations, Evolutionary Equations, vol. 2, pp. 461–559 (2005)
42.
Mielke, A., Roubíček, T.: Numerical approaches to rate-independent processes and applications in inelasticity. Math. Model. Numer. Anal. 43, 399–428 (2009)MATHCrossRef
43.
Mielke, A.: Differential, energetic, and metric formulations for rate-independent processes. Nonlinear pde’s and applications, pp. 87–170 (2011). (C.I.M.E. Summer School, Cetraro, Italy 2008, Lect. Notes Math. Vol. 2028)
44.
Mielke, A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24, 1329–1346 (2011)
45.
Mielke, A.: Emergence of rate-independent dissipation from viscous systems with wiggly energies. Contin. Mech. Thermodyn. 24(4), 591–606 (2012)
46.
Mielke, A.: Thermomechanical modeling of energy-reaction-diffusion systems, including bulkinterface interactions. Discr. Cont. Dynam. Systems Ser. S 6(2), 479–499 (2013)
47.
Mielke, A.: Deriving amplitude equations via evolutionary G-convergence. Discr. Cont. Dynam. Systems Ser. A 35(6) (2015)
48.
Mielke, A., Ortner, C., Şengül, Y.: An approach to nonlinear viscoelasticity via metric gradient flows. SIAM J. Math. Analysis 46(2), 1317–1347 (2014)
49.
Mielke, A., Peletier, M., Renger, M.: On the relation between gradient flows and the largedeviation principle, with applications to Markov chains and diffusion. Potential Anal. 41(4), 1293–1327 (2014)MATHMathSciNetCrossRef
50.
Mielke, A., Rossi, R., Savaré, G.: Modeling solutions with jumps for rate-independent systems on metric spaces. Discr. Cont. Dynam. Systems Ser. A 25(2), 585–615 (2009)MATHCrossRef
51.
Mielke, A., Rossi, R., Savaré, G.: Nonsmooth analysis of doubly nonlinear evolution equations. Calc. Var. Part. Diff. Eqns. 46(1–2), 253–310 (2013)
52.
53.
Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Springer (2015)
54.
Mielke, A., Roubíček, T., Thomas, M.: From damage to delamination in nonlinearly elastic materials at small strains. J. Elasticity 109, 235–273 (2012)MATHMathSciNetCrossRef
55.
Mielke, A., Roubíček, T., Stefanelli, U.: G-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Part. Diff. Eqns. 31, 387–416 (2008)MATHCrossRef
56.
Mielke, A., Stefanelli, U.: Linearized plasticity is the evolutionary G-limit of finite plasticity. J. Eur. Math. Soc. 15(3), 923–948 (2013)MATHMathSciNetCrossRef
57.
Mielke, A., Thomas, M.: GENERIC-a powerful tool for thermomechanical modeling. In preparation (2014)
58.
Mielke, A., Timofte, A.M.: Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39(2), 642–668 (2007)MATHMathSciNetCrossRef
59.
Mielke, A., Truskinovsky, L.: From discrete visco-elasticity to continuum rate-independent plasticity: rigorous results. Arch. Rational Mech. Anal. 203(2), 577–619 (2012)MATHMathSciNetCrossRef
60.
Negri, M.: Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim. Calc. Var. 20(4), 983–1008 (2014)
61.
62.
Onsager, L.: Reciprocal relations in irreversible processes, I+II. Phys. Rev. 37, 405–426. (part II, 38:2265–2279) (1931)
63.
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Different. Eqns. 26, 101–174 (2001)MATHCrossRef
64.
Prandtl, L.: Gedankenmodel zur kinetischen Theorie der festen Körper. Z. angew. Math. Mech. (ZAMM) 8, 85–106 (1928)
65.
66.
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)
67.
Rossi, R., Savaré, G.: Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM Control Optim. Calc. Var. 12, 564–614 (2006)
68.
Roubíček, T.: Nonlinear partial differential equations with applications. Birkhäuser Verlag, Basel (2005)
69.
Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Comm. Pure Appl. Math. LVII, 1627–1672 (2004)
70.
Savaré, G.: Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds (2011)
71.
Serfaty, S.: Gamma-convergence of gradient flows on Hilbert spaces and metric spaces and applications. Discr. Cont. Dynam. Syst. Ser. A 31(4), 1427–1451 (2011)MATHMathSciNetCrossRef
72.
Stefanelli, U.: The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47(3), 1615–1642 (2008)MATHMathSciNetCrossRef
73.
Tartar, L.: Nonlocal effects induced by homogenization. Partial Different. Eqns. Calcul. Variat. 2, 925–938 (1989)MathSciNet
74.
Tartar, L.: Memory effects and homogenization. Arch. Rational Mech. Anal. 111(2), 121–133 (1990)
Metadaten
Titel
On Evolutionary -Convergence for Gradient Systems
verfasst von
Alexander Mielke
Copyright-Jahr
2016
Buch
Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity

Print ISBN: 978-3-319-26882-8

Electronic ISBN: 978-3-319-26883-5

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-26883-5_3

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