Sobolev and BV estimates for the JKO scheme
Prof. Filippo Santambrogio, Institut Camille Jordan, Université Claude Bernard – Lyon
Sala conferenze IMATI-CNR, Pavia – Martedì 12 Marzo 2019 h.15:00
Abstract. 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 exploi (weighted) BV estimates for fast diffusion equations. The applications to various PDEs with linear diffusion, including Keller-Segel equations for chemotaxis are part of a joint ongoin work with Di Marino.