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On the heat flow on metric measure spaces: existence, uniqueness and stability

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Abstract

We prove existence and uniqueness of the gradient flow of the Entropy functional under the only assumption that the functional is λ-geodesically convex for some \({\lambda\in\mathbb {R}}\). Also, we prove a general stability result for gradient flows of geodesically convex functionals which Γ−converge to some limit functional. The stability result applies directly to the case of the Entropy functionals on compact spaces.

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References

  1. Ambrosio L., Gigli N., Savaré G.: Gradient flows in metric spaces and in spaces of probability measures. Birkäuser, Basel (2005)

    MATH  Google Scholar 

  2. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate studies in mathematics, vol. 33. American Mathematical Society, Providence, RI (2001). http://www.pdmi.ras.ru/staff/burago.html

  3. Dal Maso, G.: An introduction to Γ-convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkäuser, Boston (1993)

  4. Figalli, A., Gigli, N.: A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions (submitted paper)

  5. Fukaya K.: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87(3), 517–547 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Of progress in mathematics, vol. 152. Birkhäuser Boston Inc., Boston, MA (1999), Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by S. M. Bates

  7. Juillet, N.: Geometric inequalities and generalized Ricci bounds in the Heisenberg group. IMRN (to appear)

  8. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (to appear)

  9. Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Part. Differ. Equat. 26, 101–174 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Savaré G.: Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Math. Acad. Sci. Paris 345, 151–154 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Sturm K.-T.: On the geometry of metric measure spaces. I–II. Acta Math. 196(1), 65–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sturm, K.-T., Ohta, S.I.: Heat flow on Finster manifolds. CPAM (to appear)

  13. Villani C.: Optimal Transport, Old and New. Springer Verlag, Heidleberg (2008)

    Google Scholar 

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Authors and Affiliations

  1. University of Bordeaux 01, Bordeaux, France

    Nicola Gigli

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  1. Nicola Gigli
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Correspondence to Nicola Gigli.

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Communicated by L. Ambrosio.

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Gigli, N. On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. 39, 101–120 (2010). https://doi.org/10.1007/s00526-009-0303-9

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  • DOI: https://doi.org/10.1007/s00526-009-0303-9

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