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Global attractors for gradient flows in metric spaces. (English) Zbl 1215.35036

The authors develop the long-time analysis for gradient flow equations in metric spaces. In particular, two notions of solutions for metric gradient flows, i.e. energy and generalized solutions, are considered. For both notions of solutions the existence of the global attractor is proved. The considered notions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces, to Wasserstein spaces of probability measures. The authors present applications of their abstract results, given in the first part of the paper, by proving the existence of the global attractor for the energy solutions, both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35B41 Attractors
35K93 Quasilinear parabolic equations with mean curvature operator
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[1] Agueh, M., Asymptotic behavior for doubly degenerate parabolic equations, C. R. Math. Acad. Sci. Paris, 337, 331-336 (2003) · Zbl 1029.35144
[2] Akagi, G., Global attractors for doubly nonlinear evolution equations with non-monotone perturbations (2008), preprint
[3] Ambrosio, L., Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19, 191-246 (1995) · Zbl 0957.49029
[4] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs (2000), Clarendon Press: Clarendon Press Oxford · Zbl 0957.49001
[5] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics (2005), ETH/Birkhäuser Verlag: ETH/Birkhäuser Verlag Zürich/Basel · Zbl 1090.35002
[6] Attouch, H., Variational Convergence for Functions and Operators, Applicable Mathematics Series (1984), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, MA · Zbl 0561.49012
[7] Ball, J. M., Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7, 475-502 (1997) · Zbl 0903.58020
[8] Ball, J. M., Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10, 31-52 (2004) · Zbl 1056.37084
[9] Caraballo, T.; Marin-Rubio, P.; Robinson, J. C., A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11, 297-322 (2003) · Zbl 1053.47050
[10] Carrillo, J. A.; McCann, R. J.; Villani, C., Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19, 971-1018 (2003) · Zbl 1073.35127
[11] Carrillo, J. A.; McCann, R. J.; Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179, 217-263 (2006) · Zbl 1082.76105
[12] Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9, 428-517 (1999) · Zbl 0942.58018
[13] Chepyzhov, V. V.; Vishik, M. I., Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, vol. 49 (2002), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0986.35001
[14] 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, 180-187 (1980) · Zbl 0465.47041
[15] De Giorgi, E., New problems on minimizing movements, (Baiocchi, Claudio; Lions, Jacques Louis, Boundary Value Problems for PDE and Applications (1993), Masson: Masson Paris), 81-98 · Zbl 0851.35052
[16] Kruger, A. J.; Mordukhovich, B. S., Extremal points and the Euler equation in nonsmooth optimization problems, Dokl. Akad. Nauk BSSR, 24, 684-687 (1980), 763 · Zbl 0449.49015
[17] Jordan, R.; Kinderlehrer, D.; Otto, F., The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, 1-17 (1998) · Zbl 0915.35120
[18] Luckhaus, S., Solutions for the two-phase Stefan problem with the Gibbs-Thomson Law for the melting temperature, European J. Appl. Math., 1, 101-111 (1990) · Zbl 0734.35159
[19] Luckhaus, S.; Sturzenhecker, T., Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations, 3, 253-271 (1995) · Zbl 0821.35003
[20] Melnik, V. S.; Valero, J., On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6, 83-111 (1998) · Zbl 0915.58063
[21] Melnik, V. S.; Valero, J., On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8, 375-403 (2000) · Zbl 1063.35040
[22] Mordukhovich, B. S., Nonsmooth analysis with nonconvex generalized differentials and conjugate mappings, Dokl. Akad. Nauk BSSR, 28, 976-979 (1984) · Zbl 0557.49007
[23] Mordukhovich, B. S., Variational Analysis and Generalized Differentiation. I. Basic Theory, Grundlehren der Mathematischen Wissenschaften, vol. 330 (2006), Springer-Verlag: Springer-Verlag Berlin
[24] F. Otto, Doubly degenerate diffusion equations as steepest descent, Manuscript, 1996.; F. Otto, Doubly degenerate diffusion equations as steepest descent, Manuscript, 1996.
[25] Otto, F., Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory, Arch. Ration. Mech. Anal., 141, 63-103 (1998) · Zbl 0905.35068
[26] Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26, 101-174 (2001) · Zbl 0984.35089
[27] Rocca, E.; Schimperna, G., Universal attractor for some singular phase transition systems, Phys. D, 192, 279-307 (2004) · Zbl 1062.82015
[28] Röger, M., Existence of weak solutions for the Mullins-Sekerka flow, SIAM J. Math. Anal., 37, 291-301 (2005) · Zbl 1088.49031
[29] Rossi, R.; Mielke, A.; Savaré, G., A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7, 97-169 (2008) · Zbl 1183.35164
[30] Rossi, R.; Savaré, G., Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12, 564-614 (2006) · Zbl 1116.34048
[31] Rossi, R.; Segatti, A.; Stefanelli, U., Attractors for gradient flows of nonconvex functionals and applications, Arch. Ration. Mech. Anal., 187, 91-135 (2008) · Zbl 1151.35010
[32] Schimperna, G.; Segatti, A.; Stefanelli, U., Well-posedness and long-time behavior for a class of doubly nonlinear equations, Discrete Contin. Dyn. Syst., 18, 15-38 (2007) · Zbl 1195.35185
[33] Schimperna, G.; Segatti, A., Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations, Asymptot. Anal., 56, 61-86 (2008) · Zbl 1146.35019
[34] Segatti, A., Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst., 14, 801-820 (2006) · Zbl 1092.37052
[35] Sell, G. R., Differential equations without uniqueness and classical topological dynamics, J. Differential Equations, 14, 42-56 (1973) · Zbl 0259.54033
[36] Sell, G. R., Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8, 1-33 (1996) · Zbl 0855.35100
[37] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30 (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0207.13501
[38] Temam, R., Infinite Dimensional Mechanical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68 (1988), Springer-Verlag: Springer-Verlag New York
[39] Villani, C., Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58 (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1106.90001
[40] Villani, C., Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, vol. 338 (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1156.53003
[41] Visintin, A., Models of Phase Transitions, Progress in Nonlinear Differential Equations and Their Applications, vol. 28 (1996), Birkhäuser: Birkhäuser Boston, MA · Zbl 0882.35004
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