EYAWKA GNIPSOS & DoDeCS -- Gagliardo-Nirenberg interpolation and parabolic smoothing in Orlicz spaces and doubly degenerate chemotaxis systems

• Michael Winkler

Room: Salle Fokko du Cloux In its first part, this presentation revisits a basic question from parabolic regularity theory, and discusses some recent developments concerned with heat semigroup estimates and Gagliardo-Nirenberg interpolation involving certain Orlicz type expressions. An outcome of this is thereafter applied to a taxis-type parabolic model for the dynamics of microbial populations in nutrient-poor environments, containing some cross-degenerate diffusion mechanism as a core characteristic. The intention here is to outline an approach which, by relying on a result achieved in the first part in a crucial place, facilitates an appropriate control of such cross-degeneracies. In convex planar domains, this leads not only to a fairly comprehensive theory of global solvability, but also to a description of large time behavior and structure formation.

A Li-Yau and Aronson-Bénilan approach for the Keller--Segel system

• Alejandro Fernández Jiménez

Room: Salle Fokko du Cloux On this talk we will focus on the Keller--Segel system \begin{equation*} \begin{cases} \displaystyle\partial_t\rho = \Delta \rho^m - \mathrm{div}\left(\rho \,\nabla u \right) & \text{in } (0,\infty) \times \mathbb{R}^d,\\[6pt] -\Delta u = \rho & \text{in } (0,\infty)\times \mathbb{R}^d, \end{cases} \end{equation*} for $d \geq 2$ and $m = 2 - \frac{2}{d}$, i.e. the critical exponent. This system exhibits a rich behaviour and its dynamics depend on the initial mass. When the mass is below certain threshold (subcritical mass) there is global-in-time bounded solutions, if we are beyond this threshold (supercritical mass), one can construct solutions with finite time blow-up. Finally, if the mass is critical there exists global-in-time solutions but they are not bounded globally-in-time. The main goal of the talk is to extend the classical Li--Yau and Aronson--Bénilan estimates in order to cover the Keller--Segel case. We are able to recover the estimate for subcritical and critical mass and, in particular, for a small (computable) mass we also obtain a regularising effect. We follow two strategies: for the small mass case we rely on concavity and harmonic analysis. For the general case of subcritical and critical mass our argument is based on a careful analysis of the subsolutions of the Liouville and the Lane--Emden equations combined with a contradiction argument. The talk presents joint work with C. Elbar and F. Santambrogio.

A review of some old and more recent results on Keller-Segel and related problems

• Jean Dolbeault

Room: Salle Fokko du Cloux The goal of the lecture is to present some results on the simplest Keller-Segel model in regimes with sub-critical, critical and super-critical masses, without pretending to an exhaustive review. An emphasis will be put on problems that have been left open and related models which raise similar questions.

Li-Yau and Aronson-Bénilan Type Estimates in the JKO Scheme

• Fanch Coundreuse

Room: Salle Fokko du Cloux The Li–Yau and Aronson–Bénilan estimates are classical inequalities in the theory of the porous medium, heat, and fast diffusion equations. In this talk, I will explore how similar inequalities can be obtained at the level of Wasserstein gradient flow discretizations of these equations, namely the so-called JKO scheme. We will see that the Li–Yau estimate, in the strong Hamilton matrix inequality form, can be fully recovered in the torus and the whole space, while a version of the Aronson–Bénilan estimate holds in dimension one or two and in simple domains. This work is based on arxiv.org/abs/2510.09231 and arxiv.org/abs/2604.04169.

A finite-volume scheme for aggregation-diffusion equations with non-linear mobility

• David Gómez-Castro

Room: Salle Fokko du Cloux The aim of this talk is to discuss a finite-volume scheme for the aggregation-diffusion family of equations with non-linear mobility ∂t ρ = div (m(ρ) D (U'() + V + W*ρ)) in bounded domains with no-flux conditions. We will present basic properties of the scheme: existence, decay of a free, and comparison principle (where applicable); and a convergence-by-compactness result for the saturation case where m(0)=m(1)= 0, under general assumptions on m, U, V, and W. The results are joint works published with J.-A. Carrillo and A. Fernandez-Jimenez. At the end of the talk, we will discuss an extension to the Porous-Medium Equation with non-local pressure that corresponds to m(ρ) = ρ^m, U, V = 0 and W(x) = c|x|^{-d-2s}.

Quantitative convergence of moderately interacting particle systems with singular kernels

• Milica Tomasevic

Room: Salle Fokko du Cloux In this talk, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. Using a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting non-linear Fokker-Planck equation. The convergence rates are polynomial and involve, on one side the regularity parameters of the interaction kernel, and on the other side, the stochastic convolution integral decay rate. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos). These results only require very weak regularity on the interaction kernel, which permits to treat models for which the classical particle system is not known to be well-defined. For instance, this includes attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. We weill tackle also some recent advances, including Burgers equation and the case of non-conservative Keller-Segel model with logistic source (involving a branching and interacting particle system). Based on works with A. Richard (CentraleSupelec), C. Olivera (Unicamp) and T. Cavallazzi.

Mean-field derivation of signal-dependent Keller-Segel system

• Li Chen

Room: Salle Fokko du Cloux In this talk, I will present a mean-field derivation of the signal-dependent Keller-Segel system through moderate stochastic particle system in the two-dimensional whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals. The convergence is proved by introducing an intermediate particle system with a mollified interaction potential. We present the propagation of chaos result with two different types of mollification scaling, namely logarithmic and algebraic scaling.

Global existence for the two-dimensional fully parabolic Keller–Segel system at critical mass

• Tatsuya Hosono

Room: Salle Fokko du Cloux We consider the Cauchy problem for the two-dimensional fully parabolic classical Keller–Segel system. Since its introduction in the 1970s, this system has been extensively studied from various perspectives. One of the main topics is the so-called critical mass phenomenon, namely the $L1$ -threshold behavior in two spatial dimensions. Global existence at the critical mass has been established under radial symmetry, additional moment conditions on the initial data, or for the parabolic-elliptic system as a simplified model. These restrictions arise because classical energy-based methods lose effectiveness at the critical level, and controlling the behavior of solutions at spatial infinity becomes a major difficulty. However, such assumptions are extrinsic to the intrinsic scaling structure underlying the critical mass phenomenon. In this talk, we establish global-in-time existence at the critical mass for general initial data, without any additional assumptions. The proof is based on a refined Lyapunov functional and associated dissipative estimates, which allow us to recover sufficient control of the dynamics in the whole space.

On patterns of singularity formation for the parabolic-elliptic Keller-Segel system

• Charles Collot

Room: Salle Fokko du Cloux he parabolic-elliptic Keller-Segel system models cell motion under chemotaxis. It is a mass preserving equation that has the same scaling invariance as the quadratic semilinear heat equation. It admits blowup solutions in the mass critical two-dimensional case as well as in higher dimensions that are mass supercritical. In such instances when the density becomes singular in finite time, this describes cell aggregation. This talk will first review four previously known blow-up patterns (self-similar, flat, collapsing steady state, collapsing sphere). It will then present a new one in the mass critical case: where two stationary states are simultaneously collapsing and colliding at a single singular point. A formal blow-up law was proposed by Herrero-Seki-Velazquez in 2014. We provide a rigorous construction of such solution. We will explain some of the new ideas to study this dynamics that to our knowledge had not been studied before in evolution pdes, where two solitons interact in the same parabolic neighborhood from the singularity in a non-radial configuration, together with the radiation remainder. This is joint work with T.-E. Ghoul (NYU Abu Dhabi), N. Masmoudi (NYU Abu Dhabi and Courant Institute) and V. T. Nguyen (National Taiwan University).

Radial solutions of the minimal chemotaxis model in R^d

• Piotr Biler

Room: Salle Fokko du Cloux We discuss existence of radially symmetric solutions (evolution and self-similar cases) of the minimal Keller-Segel system in $\mathbb R^d$: $$u_t=\Delta u- \nabla\cdot(u\nabla v),$$ $$\Delta v+u=0,$$ under optimal assumptions on the initial data u0 = u(0; :). We are interested, in particular, in minimal regularity assumptions imposed on the initial data in order to a local-in-time solution does exist, as well as size conditions for (approximate) dichotomy: global-in-time existence versus finite time blowup of solutions.

A degenerate chemotaxis system with indirect signal production

• Philippe Laurençot

Room: Salle Fokko du Cloux Global existence of weak solutions to a fully parabolic chemotaxis system with indirect signal production and degenerate cell diffusion is shown. The proof is based on a discrete time scheme introduced by Yoshifumi Mimura (2024), for which each equation can be solved separately by a suitable variational approach. Maximal regularity at the discrete level also plays an important role (on-going joint work with Tatsuya Hosono, Osaka).