Invited advanced seminar: Some PDEs with competition effects and functional inequalities

Tuesdays 8:30-11:45 (from 21.11.2017 to 20.12.2017)

The seminar is given by: Prof. Jinhuan Wang

Abstract: In many physical and biological systems, there are some competing effects such as focus and de-focus, attraction and repulsion, spread and concentration. These competing effects usually are represented by terms with different signs in a free energy. The dynamics of the physical system sometimes can be described by a gradient flow driven by the free energy. Some functional inequalities can be used to determine the domination among these competing effects in the free energy, and provided sharp conditions on initial data or coefficients in the system for the global existence.

In these seminars, we will introduce two kinds of Partial differential equations (PDEs) with competing effects: Keller-Segel equations and Thin film equation. We discuss properties of solutions to the two kinds of PDEs under sharp initial conditions. Moreover, we will reveal some important relations between functional inequalities and sharp conditions for the global existence to seme PDEs. For example, the Hardy-Littlewood-Sobolev inequality vs parabolic-elliptic Keller-Segel model, Onofri's inequality vs parabolic-parabolic Keller-Segel model, and Sz. Nagy inequality vs 1-D thin film equation etc.. Finally, we will talk about the best constant for Gagliardo-Nirenberg interpolation inequalities.