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Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

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).

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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.

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Correspondence to Eva Kopfer.

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Communicated by L. Ambrosio.

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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

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  • DOI: https://doi.org/10.1007/s00526-017-1287-5

Mathematics Subject Classification

  • 35K05
  • 35K08
  • 49J40
  • 49J52
  • 58J35