Entropy Methods and Nonlinear Diffusions
speaker: Jean Dolbeault (Université Paris Dauphine)abstract: Entropy methods have been largely studied during recent years. In these lectures, the emphasis will be put on how to relate optimal rates for large time behavior of solutions to diffusion equations in the whole euclidean space with optimal constants in functional inequalities. The following topics will be covered:
(1) Linear Fokker-Planck equation, generalized entropies and gradient flow structure for various notions of distances
(2) Porous media and fast diffusion equations: global decay rates towards Barenblatt profiles can be related to Gagliardo-Nirenberg interpolation inequalities in a certain range of parameters. Outside of this range, a linearized version of the inequalities still holds, the so-called Hardy-Poincaré inequalities, which govern the asymptotic rates of decay.
(3) Best matching Barenblatt profiles provide refined asymptotic expansions of the solutions. Such an approach has rich consequences for underlying functional inequalities and provide refined versions of the Gagliardo-Nirenberg inequalities, including in the limiting case of Sobolev's inequality.
(4) Relative entropy methods also apply to systems like the parabolic-elliptic Keller-Segel model. Various inequalities related to the optimal logarithmic Hardy-Littlewood-Sobolev inequality will be considered in view of a detailed description of the asymptotic behavior of the solutions.
timetable:
Tue 26 Jun, 11:00 - 12:30, Aula Dini
Wed 27 Jun, 9:00 - 10:30, Aula Dini
Thu 28 Jun, 15:30 - 17:00, Aula Dini
Fri 29 Jun, 9:00 - 10:30, Aula Dini
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