Calculus of Variations and Geometric Measure Theory

Sobolev and BV estimates for the JKO scheme

Filippo Santambrogio (Institut Camille Jordan, Université Claude Bernard - Lyon 1)

created by alberti on 08 Apr 2019

15 apr 2019 -- 14:45   [open in google calendar]

Sala Riunioni del Dipartimento di Matematica, Università di Pisa,


I will show which kind of uniform BV and Sobolev estimates can be obtained for some equations which are gradient flows in the Wasserstein spaces and which range from Fokker-Planck to Keller-Segel systems or nonlinear diffusion. This will be based on optimal transport tools applied to the Jordan-Kinderlehrer-Otto scheme, using in particular on a new inequality (five-gradients-inequality) that we recently found in collaboration with De Philippis, Mészáros and Velichkov, in a work where we also provide an easy BV estimate for porous-medium--type diffusion. Similarly, in a recent work with Iacobelli and Patacchini we obtain and exploit (weighted) BV estimates for fast diffusion equations. The applications to various PDEs with linear diffusion, including Keller-Segel equations for chemiotaxis are part of a joint ongoing work with Di Marino.