Renjun Duan - The Chinese University of Hong Kong
Kinetic shear flow via the Boltzmann equation
In the talk we first provide a survey of recent results on the kinetic shear flow governed by the nonlinear Boltzmann equation in both spatially homogeneous and inhomogeneous settings. Then, we focus on a specific problem on the time-evolutionary 3D kinetic Couette flow under the diffusive limit, in particular, showing the convergence to the 1D steady kinetic Couette flow uniformly in large time and small Knudsen number.
Giacomo Albi - University of Verona
Kinetic modelling of collective opinion dynamics with social heterogeneity
In this talk we discuss kinetic models for opinion formation in heterogeneous populations. First, we consider social heterogeneity through network connectivity, where highly connected agents exert stronger influence, and derive the corresponding kinetic description from microscopic interactions. Secondly, we introduce payoff-driven adaptive strategies, coupling public opinion, private conviction, and behavioural response. A Boltzmann model and its asymptotic behaviours are derived. While strategy heterogeneity generally prevents macroscopic closure, state-independent payoffs yield an exact finite-dimensional parametrization of the strategy distribution. We finally analyze fast-opinion and fast-strategy regimes, characterizing reduced dynamics, stationary states, and the emergence of compromise, stubbornness, and mixed behaviours. Numerical experiments complement the analysis illustrating different dynamical regimes, and the interplay between social heterogeneity, strategic adaptation, and opinion formation.
Katarzyna Ryszewska - Institute of Mathematics of Polish Academy of Sciences
An application of the mean-field limit to non-exchangeable non-conservative systems with adaptive weights
In this talk, we briefly introduce the concept of the mean-field limit for large particle systems. We review the novel approach introduced by Jabin, Poyato, and Soler (2025), which employs extended graphons to establish the mean-field limit for non-exchangeable agents under mild assumptions on the interaction matrix. Notably, this framework accommodates sparse connection matrices. Subsequently, we present the attempt to apply this result to cover some non-conservative problems. Finally, we introduce the notion of vector-valued dynamic extended graphon, which allows to tackle certain class of problems with adaptive dynamic weights with possibly sparse connections. This is a joint work with Piotr Gwiazda.
Víctor Villegas-Morral - Universitat Politecnica de Catalunya
Decision-making in heterogeneous self-propelled systems across multiple scales
Inspired by empirical observations in animal swarming -- particularly in schooling fish -- we propose an opinion-swarming model for self-propelled particles in order to understand the effect of uninformed individuals in consensus formation. Building on classical bounded-confidence opinion models and self-propelled swarming models, we introduce a three-population framework that distinguishes between leaders, followers, and uninformed individuals. Particles are described by their position, velocity, and a continuous opinion variable; they interact through self-propulsion, alignment, attraction-repulsion forces, and opinion-based mechanisms. First, we derive and analyse an individual-based coupled model that integrates spatial swarming dynamics with the evolution of individual opinions. We perform an extensive numerical study of the relevant parameters and their effect on the dynamics. We show the formal and rigorous derivation of the mean-field partial differential equations for this coupled individual-based model, which lays the basis for the study of the long-term behaviour of the system. Our analysis reveals that uninformed individuals, despite lacking any opinion bias, significantly influence group dynamics by diluting the effect of leaders and promoting more democratic decision-making. These findings support the role of uninformed agents in collective decision-making and provide the first analytical insights into leadership and decision-making in heterogeneous crowds.
Junhao Zhang - University of Warsaw
Global Dynamics of Compressible Navier–Stokes–Riesz Systems
The compressible Navier–Stokes–Riesz system describes viscous compressible fluids subject to a nonlocal self-consistent force. Depending on the sign of the coupling, the Riesz interaction may be attractive or repulsive.
We first discuss the attractive case. For a one-dimensional physical-vacuum free-boundary problem, we prove global existence of strong solutions and asymptotic stability of compactly supported steady states using weighted energy estimates and the Kazhikov–Shelukhin type representation formula. We then turn to the repulsive Cauchy problem in $\mathbb{R}^{3}$. Combining dispersive estimates, normal-form analysis, and nonlinear energy methods in both positive and negative Sobolev spaces, we establish global existence of smooth solutions, $L^{2}–L^{\infty}$ decay, and a global-in-time inviscid limit to an irrotational Euler–Riesz flow. Since the $L^{2}$ decay obtained in this inviscid-limit framework is not sharp, we further investigate optimal decay rates within the same framework.
Ansgar Jüngel - Vienna University of Technology
Nonlocal cross-diffusion equations for segregating populations
Cross-diffusion equations play a fundamental role in the mathematical description of interacting and segregating population species, with the Busenberg-Travis and Shigesada-Kawasaki-Teramoto (SKT) models as prominent examples. In this talk, we discuss two nonlocal extensions of these models. The first model consists of Busenberg-Travis-type equations, in which Darcy's law is replaced by the elliptic Brinkman law. The latter accounts for viscous stresses, thus introducing nonlocality into the velocity field. The second model is a nonlocal extension of a SKT model, in which the Laplacian is replaced by a nonlocal difference integral operator. For both models, we establish the global-in-time existence of nonnegative weak solutions using entropy methods and investigate conditions under which the solutions remain bounded. The analysis of the nonlocal SKT model requires novel discrete chain-rule inequalities. Beyond their analytical role, these inequalities are also useful in the construction of entropy-dissipating finite-volume discretizations.
Stefano Spirito - University of L’Aquila
Global Weak Solutions for Korteweg-Type Fluid Models
Abstract. We present recent results on the global-in-time existence of finite-energy weak solutions for one-dimensional compressible fluid models with Korteweg-type capillarity and density-dependent, possibly highly degenerate, viscosity coefficients. The analysis relies on the interplay between energy estimates, BD-type entropy structures, nonlinear coercivity properties, and compactness arguments based on suitable truncations. If time permits, we will also discuss connections with the gradient-flow structure of the Korteweg energy.
Oliver Tse - Eindhoven University of Technology
Gradient Structures and Mean-Field Limits in Population Dynamics
Particle systems in population dynamics—such as variations of the Bolker–Pacala–Dieckmann-Law model—are governed by a forward Kolmogorov equation for measure-valued processes. I'll show how to construct a generalized gradient structure for this equation that incorporates fluxes from birth and death events.
In the large population limit, this equation converges to a Liouville transport equation governing the mean-field dynamics. The gradient structures themselves converge in the sense of Energy-Dissipation Principles. This convergence establishes propagation of chaos for the particle system and yields a gradient-flow formulation for the mean-field limit—connecting microscopic particle dynamics to macroscopic continuum behavior through gradient structures.
David Poyato - University of Granada
Mean field limit of non exchangeable interacting diffusions on co-evolutionary networks
Traditional models of interacting particle systems often assume a fixed network of connections, which simplifies the analysis but fails to capture many real-world phenomena. Indeed, interacting particles where the network structure and particle states co-evolve in mutual influence, are increasingly recognised as essential in diverse fields. For instance, they appear in neuroscience, where learning is encoded through the strengthening and weakening of synaptic connections. In this talk I will present the rigorous mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The main challenge arises from the coupling between the network dynamics and the agents` states, which results in a non-Markovian dynamics where the system`s future depends on its entire history. Consequently, the mean-field limit is not described by a partial differential equation, but by a system of non-Markovian stochastic integrodifferential equations. A second difficulty stems from the non-linear weight dynamics, which requires a careful choice for the limiting network structure. Due to the limitations of the classical theory of graphons (Lov\`asz and Szegedy, 2006), in our mean-field limit we employ for the first time K-graphons (Lov\`asz and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025), as they provide a natural framework compatible with non-linear structures. This is a joint work with Juli\'an Cabrera-Nyst (University of Granada).
Nilasis Chaudhuri - University of Warsaw
Generalized solution and Weak-Strong uniqueness for a barotropic Euler-Riesz system
We study the compressible Euler--Riesz system on the torus $\mathbb{T}^d$ ($d=2,3$). Using the Caffarelli--Silvestre extension, we replace the nonlocal repulsive interaction with a local stress tensor in an extra dimension. Leveraging energy coercivity, we introduce global-in-time dissipative solutions for arbitrary finite-energy data and prove weak--strong uniqueness via a tailored relative energy method. This result holds for all fractional orders $\beta \in (0,2)$ and adiabatic exponents $\gamma > 1$.
Tomasz Dębiec - University of Warsaw
Singular limits arising in a two-species tissue-growth model with nonlocal Brinkman coupling
We investigate a two-species advection-reaction system modelling the growth of living tissues. The cell densities are advected by the gradient of a chemical potential which satisfies the so-called Brinkman law, while the growth rate of each population is governed by a function of the joint population pressure. I will discuss recent results regarding the behaviour of solutions to this system as certain model parameters reach critical values: the “inviscid limit” as the viscosity in the velocity law vanishes, and the “incompressible limit” as the stiffness of the pressure-density law becomes infinite.
Juan Soler - University of Granada
Geometric Coherence for Vortex Patch Contours and V-states: The Arc+Gap Principle at the Zygmund Endpoint
We study the short-time evolution of planar vortex patches whose boundaries have finitely many corners. Our goal is to identify geometric conditions under which the corner structure can be continued without relying on curvature estimates.
Working in a constant-speed parametrization, we control the length of the contour and obtain a uniform bound on the tangent. A finite-direction arc+gap condition then provides an endpoint relation between the $L^\infty$ and $BMO$ size of the tangent, while Dini regularity is required only locally, away from the corners and for suitably renormalized quantities near them. Under an admissible bilateral profile at each corner, the boundary remains chord--arc for a time independent of the regularization, and the renormalized corner angles satisfy an effective evolution law.
For regular $n$-gons, we construct a dihedrally symmetric corner profile and verify the hypotheses of the abstract continuation theorem. We also discuss the related question of the existence of rotating V-states with corners in this setting.
Piotr Gwiazda - Institute of Mathematics of Polish Academy of Sciences
From Nonlocal to Local via a \(W_2\)-Stability Estimate
I will discuss a stability estimate in the 2-Wasserstein distance for two nonlinear continuity equations. When one of the equations has a Wasserstein gradient-flow structure generated by a \lambda-geodesically convex energy, the estimate controls the distance between the two solutions by the difference of the corresponding velocities evaluated at the same density. I will focus on the proof: the differential formula for {W_2}^2 is combined with geodesic convexity along the optimal-transport geodesic, endpoint first-variation inequalities, and a final Cauchy–Schwarz argument. A crucial feature is that the resulting inequality is written for W_2 itself rather than for {W_2}^2.
I will then explain how this estimate applies to the one-dimensional nonlocal approximation of the quadratic porous medium equation. It reduces the nonlocal-to-local comparison to a weighted L^2 estimate for a convolution defect of the local solution. Using gradient bounds, an Aronson–Bénilan-type estimate, and weighted second-derivative control for the local equation, the second moment of the rescaled kernel yields an O(\epsilon) convergence rate in W_2, improving the previously known O(\sqrt{\eplsion}) rate.
Joint work with José A. Carrillo and Jakub Skrzeczkowski.
Andrew Nugent - University College London
Connecting models of opinion formation across scales
This talk will introduce and connect three approaches to modelling opinion formation as an interacting particle system. We begin with an agent-based model (ABM) with a finite population and random pairwise interactions. From this we derive a system of coupled ordinary or stochastic differential equations by simultaneously rescaling time and the size of interactions, showing how the precise form of these limiting equations connects to choices made in the ABM. Building on this, the remainder of the talk focuses on the effect of adding age structure to the population. Each individual ages continuously in time until a maximum age, at which point they die and re-enter the population at age zero with a new, randomly selected opinion. We study the corresponding mean-field limit, a nonlinear, nonlocal partial differential equation, showing the existence of steady states and new complex dynamics made possible by the continuous introduction of new individuals to the population. The talk will conclude with several open questions concerning stability and the existence of periodic solutions.
Jan Peszek - University of Warsaw
Transformers as Interacting Particle Systems: Clustering, Oscillations, and Bifurcations
Transformers are the fundamental architecture behind modern large language models, but their self-attention mechanism can also be viewed as a system of particles interacting through nonlocal forces. I will begin by introducing the main ideas behind transformers, focusing on building intuition for how self-attention incorporates context into token representations and how this leads to an interacting-particle description.
I will then discuss the resulting dynamics for deep linear transformers with two-dimensional token embeddings. Through a reduction to a lower-dimensional invariant manifold, I will explain the connection with Kuramoto-type models and explore behaviors beyond consensus and clustering, including persistent oscillations and bifurcations.
Nuno Alves - King Abdullah University of Science and Technology
Upper and Lower Bounds for Riesz Interactions
Motivated by the problem of obtaining integrability estimates for the density in Euler–Riesz systems, we will discuss upper and lower bounds for Riesz interactions. First, we will describe a stress representation of the interaction force and uniform estimates for the associated bilinear fractional integrals. We will then focus on lower bounds for the Riesz energy and for a determinant functional arising from compensated integrability. We will explain the role of confinement to a fixed ball, quasi-concavity, and log-concavity in these bounds, as well as their sharpness.The energy estimates provide a geometric lower-bound counterpart to the classical Hardy–Littlewood–Sobolev inequality.
This is based on joint work with L. Grafakos and A. E. Tzavaras.