Abstract
We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time-dependent energy functionals in both settings. In particular, in the case when each underlying space satisfies a lower Ricci curvature bound in the sense of Lott, Sturm and Villani, we provide time-discrete approximations of the time-dependent heat flows introduced in Kopfer and Sturm (Heat flows on time-dependent metric measure spaces and super-Ricci flows, 2017. arXiv:1611.02570).
Similar content being viewed by others
References
Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Piccoli, B., Rascle, M. (eds.) Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics, vol. 2062, pp. 1–155. Springer, Heidelberg (2013)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probabiliy Measures. Birkhäuser, Basel (2005)
Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195(2), 289–391 (2013)
Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405–1490 (2014)
Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear Diffusion Equations and Curvature Conditions in Metric Spaces (2015). arXiv:1509.07273
Bogachev, V.I.: Measure Theory, vol. 1. Springer, Berlin (2007)
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)
Erbar, M.: The heat equation on manifolds as a gradient flow in the Wasserstein space. Annales de l’I. H. P. Probabilités et Statistiques 46(1), 1–23 (2010)
Erbar, M., Kuwada, K., Sturm, K.-T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)
Ferreira, L., Valencia-Guevara, J.: Gradient Flows of Time-Dependent Functionals in Metric Spaces and Applications for PDE (2015). arXiv:1509.0416161v1
Gigli, N.: On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. 39(1–2), 101–120 (2010)
Gigli, N., Kuwada, K., Ohta, S.: Heat flow on Alexandrov spaces. Commun. Pure Appl. Math. 66(3), 307–331 (2013)
Haslhofer, R., Naber, A.: Weak Solutions for the Ricci Flow I (2015). arXiv:1504.00911
Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)
Kopfer, E., Sturm, K.-T.: Heat Flows on Time-Dependent Metric Measure Spaces and Super-Ricci Flows (2017). arXiv:1611.02570
Lierl, J., Saloff-Coste, L.: Parabolic Harnack Inequality for Time-Dependent Non-symmetric Dirichlet Forms (2012). arXiv:1205.6493
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009)
McCann, R., Topping, P.: Ricci flow, entropy and optimal transportation. Am. J. Math. 132(3), 711–730 (2010)
Ohta, S., Sturm, K.-T.: Heat flow on Finsler manifolds. Commun. Pure Appl. Math. 62(11), 1386–1433 (2009)
Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, vol. 13. Springer, New York (2004)
Rossi, R., Mielke, A., Savaré, G.: A metric approach to a class of doubly nonlinear evolution equations and applications. Ann. Sc. Norm. Super. Pisa. 7, 97–169 (2008)
Rossi, R., Savaré, G.: Gradient flows of non convex functionals in Hilbert spaces. ESAIM Control Optim. Calc. Var. 12, 564–614 (2006)
Sturm, K.-T.: On the geometry of metric measure spaces. Acta Math. 169(1), 65–131 (2006)
Sturm, K.-T.: Super Ricci Flows for Metric Measure Spaces. I (2016). arXiv:1603.02193
Topping, P.: Lectures on the Ricci Flow, vol. 325. Cambridge University Press, Cambridge (2006)
Villani, C.: Optimal Transport, Old and New. Springer, Berlin (2009)
von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)
Acknowledgements
I would like to thank my supervisor Karl-Theodor Sturm for supporting me. I also thank Peter Gladbach for helpful comments and suggestions on this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
Cite this article
Kopfer, E. Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces. Calc. Var. 57, 20 (2018). https://doi.org/10.1007/s00526-017-1287-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-017-1287-5