Antonio Esposito: Nonlinear Degenerate Cross Diffusion Systems with Nonlocal Interaction

Explore a lecture on nonlinear degenerate cross diffusion systems with nonlocal interaction, presented by Antonio Esposito at the Hausdorff Center for Mathematics. Delve into a joint work investigating partial differential equations with nonlinear cross-diffusion and nonlocal interactions, applicable to social sciences, finance, biology, and real-world scenarios. Learn about the global-in-time existence of weak solutions using a semi-implicit version of the Jordan-Kinderlehrer-Otto scheme, which allows for consideration of nonlocal interaction terms without a formal gradient flow structure. Examine the uniform 'coerciveness' assumption on diffusion, enabling the study of systems with degenerate cross-diffusion. Follow the lecture's structure, covering introduction, multiple species settings, examples, assumptions on interaction potentials, the semi-implicit JKO approach, piecewise constant interpolation, and flow interchange, concluding with key insights into this complex mathematical topic.

Introduction
Nonlinear diffusion + nonlocal interactions
Many species
Setting
Examples of A
Assumptions on the interaction potentials
Goal: Existence of weak solutions
Semi-Implicit JKO: our case
Piecewise constant interpolation
Flow Interchange
Conclusion