Lecturers

Young-Pil Choi, Department of Mathematics, Yonsei University, Seoul, Republic of Korea

Collective dynamics: from particle systems to continuum models

Collective behaviors such as aggregation, flocking, and synchronization arise in a wide range of biological and social systems, including bird flocks, fish schools, human crowds, and opinion dynamics. A common mathematical approach to these phenomena begins with microscopic particle systems in which individual agents interact through simple local or nonlocal rules, such as attraction, repulsion, and velocity alignment. Despite the simplicity of these interaction mechanisms, the resulting systems may exhibit highly organized large-scale patterns.

In this lecture series, we will introduce several prototypical models of collective dynamics and examine how their mathematical structures describe the transition from disordered motion to coordinated behavior. Particular attention will be given to interacting particle systems for aggregation and flocking and to the qualitative properties that govern their long-time behavior. When the number of agents becomes large, a continuum description becomes both mathematically natural and computationally advantageous. We will thus discuss the passage from microscopic particle models to mesoscopic kinetic equations and macroscopic hydrodynamic models. Examples include kinetic alignment equations, pressureless Euler systems with nonlocal interactions, and aggregation--diffusion equations. We will review mathematical techniques used in the rigorous derivation and analysis of these continuum models, with emphasis on compactness methods, Wasserstein stability estimates, and modulated energy arguments.

The lectures will combine modeling perspectives with rigorous analytical methods and are intended to provide a coherent introduction to the connections among interacting particle systems, kinetic equations, and hydrodynamic equations. They are aimed at graduate students and researchers interested in kinetic theory, fluid mechanics, nonlocal PDEs, and the mathematics of complex systems.

Eduard Feireisl, Institute of Mathematics, Academy of Science of Czech Rep., Prague, Czech Rep.

The Euler system of gas dynamics

This is a survey lecture series highlighting several recent results concerning well/ill posedness of the Euler system of gas dynamics. Solutions of the system are identied as limits of consistent approximations generated either by physically more complex problems, notably the Navier-Stokes-Fourier system, or by the approximate schemes in numerical experiments. The role of the fundamental principles encoded in the First and Second law of thermodynamics in identifying a unique physically admissible solution is examined.

Jakub Skrzeczkowski,  Mathematical Institute, University of Oxford, Oxford, United Kingdom

Compensated Compactness Methods and Cross-Diffusion Systems

This four-lecture mini-course provides a rigorous overview of contemporary compensated compactness techniques and their application to non-linear parabolic systems, with a primary focus on cross-diffusion models arising in mathematical biology. The course is organized into two parts:

Part I: Compactness techniques (Lectures 1–2) The first part introduces several tools for passing to the limit in non-linear PDEs. We begin by contrasting weak versus strong compactness, detailing why classical functional analytic methods often fail in non-linear settings. We introduce Young measures to characterize fine oscillations and concentration effects. Furthermore, we recall the Div-Curl Lemma and related compensated compactness framework, illustrating how algebraic structural properties allow one to identify weak limits of non-linear products. Simple, concrete examples, including applications to basic models in mathematical biology, will be discussed throughout to build intuition.

Part II: Cross-diffusion system with different advection (Lectures 3–4) The second part applies these analytical techniques to a cross-diffusion system modeling the evolution of two biological species subject to different advection fields. The existence of solutions has been open for a long time (in fact, it is still open in an arbitrary number of dimensions). Here, we discuss a recent work establishing the existence of solutions in dimension d=1, using compensated compactness techniques. We begin with the mathematical motivation and derivation of such systems, highlighting how independent advection fields break structural symmetries and create non-trivial mathematical obstacles. Then, we will detail a proof for global existence under a quadratic pressure law following the compensated compactness approach in arXiv:2603.20153. Finally, if time permits, we will briefly discuss the recent work of Elbar on the system with different mobilities (arXiv:2604.14775).

Endre Süli, Mathematical Institute, University of Oxford, Oxford, United Kingdom

Navier–Stokes–Fokker–Planck–Systems

Non-Newtonian fluids arise in a number of industrial applications, and their mathematical modeling is therefore of significant practical importance. Independently, mathematical models of non-Newtonian fluids give rise to interesting and challenging analytical problems, which have been the driving force behind a large body of research in the field of PDE theory over the last two decades, aimed at understanding the well-posedness of various models proposed in the non-Newtonian fluid mechanics and polymer physics literature.

Since the pioneering research of W. Kuhn, H.A. Kramers, P.-G. De Gennes, M. Doi, S. Edwards, and other scientists working at the interface of polymer chemistry/physics and statistical physics, kinetic models have been widely and successfully used to describe the motion of polymeric fluids. In this lecture series, we shall concentrate on a specific class of non-Newtonian fluids, dilute polymers, the underlying assumption in their mathematical description being that polymer molecules which are suspended in a (typically) Newtonian (i.e. Navier--Stokes) fluid, the solvent, exhibit no self-interaction and do not interact with each other. The associated mathematical model involves the coupling of the Fokker--Planck equation, which describes the evolution of the probability density function of the random variable satisfying a Langevin equation whose purpose is to capture the random motion of the polymer molecules in the flowing solvent, with the Navier--Stokes equations, modelling the evolution of macroscopic features of the polymeric fluid.

The lectures will focus on the derivation of the model, the question of existence of global-in-time large-data weak solutions to the resulting macro-micro Navier--Stokes--Fokker--Planck system in the isothermal and nonisothermal settings, and the problem of rigorous macroscopic closure. We shall also comment on computational challenges associated with the fact that the Fokker--Planck equation is a high-dimensional time-dependent partial differential equation, and standard numerical methods therefore suffer from the curse of dimensionality.