New Trends in Nonlinear Diffusions

Overview

The research area of nonlinear PDEs, or partial differential equations, is nowadays undergoing constant development, both due to its connections to the physical and natural phenomena they model, and due to its growing interactions with other branches of mathematics, such as geometric analysis, functional analysis, and optimal transport.

This three-week program on New Trends in Nonlinear Diffusions will feature two one-week workshops, together with a tutorial week consisting of mini-courses for junior researchers.

One of the main topics of the program is the analysis of free boundary problems. This is one of the central directions in the field of nonlinear diffusions, with applications in fluid dynamics, material science, finance, and biology. These problems involve partial differential equations where the boundary of the domain is unknown and must be determined as part of the solution. Their study combines traditional PDE techniques with ideas from harmonic analysis, probability, and geometric analysis. The program will emphasize questions concerning semi-linear equations, nonlocal problems, blow-up analysis of singular points, classification of global solutions, and generic regularity of free boundaries.

The second main topic is nonlinear diffusion equations and their connections to nonlocal interaction models. These equations have a long history in physics, and nonlinear diffusion can often be rigorously derived from stochastic interacting many-particle systems in the mean-field limit. Nonlocal interaction models also arise in kinetic theory, fluid dynamics, and biological models describing collective behavior, chemotaxis, and cross-diffusion. Mathematically, their study has led to significant developments in functional inequalities, gradient flows, optimal transport, and entropy methods.

The program aims to foster new collaborations by strengthening existing ties among researchers working in related areas, while also creating opportunities for experts in different nonlinear PDEs to exchange ideas and techniques. It will bring together leading experts in nonlinear diffusion models, free boundary problems, nonlocal many-particle systems, and kinetic PDEs, along with promising young researchers.