UNSTABLE : UNDERSTANDING THE PARABOLIC-PARABOLIC KELLER-SEGEL

12 nov. 2026, 09:00 → 13 nov. 2026, 16:30 Europe/Paris

Sorbonne Université (Salle 322, tours 15-25, 3eme étage) / INRIA Paris (Salle Anita Borg)

The aim for this workshop is twofold: to bring together people with expertise on the subject of chemotaxis and to foster discussions and interactions among participants from different fields. 

On november 12, the workshop takes place in Sorbonne Université, 4 Place Jussieu, 75005 Paris

On november 13, the workshop takes place in INRIA PARIS, 48 Rue Barrault, 75013 Paris 

Invited Speakers: 

• Monica Musso (University of Bath)
• Philippe Souplet (Université Sorbonne Paris Nord)
• Charles Collot (Cergy Paris Université)
• Maxime Herda (Inria - Lille)
• Arianne Trescase (Université Paul Sabatier)
• Milica Tomasevic (CNRS - École Polytechnique)

 

Registration is free but mandatory.

Organizers:

• Nikita Simonov (LJLL - Sorbonne Université)
• Lucilla Corrias (LaMME - Université Évry Paris-Saclay)
• Luca Alasio (LJLL-Inria)
• Nastassia Pouradier Duteil (LJLL-Inria)

 

jeu. 12 novembre

1 Accueil

2 Monica Musso: Finite-time blow-up for the three-dimensional axially symmetric Keller-Segel system

We construct axially symmetric finite-time blow-up solutions to the three-dimensional Keller–Segel system. By adapting gluing techniques, we derive a precise asymptotic expansion for type II singularities. In our construction the mass concentrates along multiple rings and we obtain a refined expansion for the blow-up rate. This work is in collaboration with Federico Buseghin, Juan Davila and Manuel del Pino.

3 Milica Tomasevic: Interacting particle systems for the doubly parabolic Keller Segel model

In this talk, I will present a series of works on the approximation of the doubly parabolic Keller-Segel model using stochastic interacting particle systems in various spatial dimensions. We will introduce two types of approximations. The first, based on the Duhamel formulation of the system, is a singularly interacting non-Markovian particle system that approximates cell positions, from which we can recover the concentration field. The second is a two-species particle system that simultaneously tracks cell positions and chemo-attractant particles (in the spirit of Stevens, 2000).

After a brief review of the 1D setting, we focus on the 2D case. We obtain the convergence of both systems to the macroscopic limit equation as the number of particles tends to infinity, provided the sensitivity parameter is sufficiently small. We will discuss also results showing that the two-species system converges to the non-Markovian system when accelerating the production rate of the chemo-attractant while keeping the number of cell particles fixed. The proofs rely in part on a novel functional inequality applied to particle trajectories.

This talk is based on joint works with N. Fournier, J.-F. Jabir, and D. Talay.

11:30 Coffee & Discussion

12:30 Lunch

ven. 13 novembre

4 Ariane Trescases: The Keller-Segel system with local sensing: global existence and long-time behaviour

We consider a class of Keller-Segel systems under Laplacian form, which models chemotactic aggregation with local sensing. We will see that while reminiscent of the classical (minimal) Keller-Segel system, which may exhibit blow-up in finite time, this class of system typically possesses global-in-time solutions. We discuss its rich long-time behaviour and, if time permits, possible extensions in presence of nonlinear self-diffusion.

10:15 Coffee & Discussions

5 Maxime Herda : Finite volumes and discrete Trudinger inequality for the Keller-Segel system with local sensing

In this talk, I will present a numerical scheme for a parabolic system which models chemotaxis with local sensing in bounded domains. This system has a similar entropy structure as the celebrated Keller-Segel system, but unlike the latter, its solutions are known to exist globally in 2D. The long-time behavior of solutions is only partially understood which motivates numerical exploration with reliable (convergent, stable) numerical methods.

In a first part I will present the linearly implicit finite volume scheme designed for the system, as well as its numerical analysis (entropic stability, convergence, asymptotic preserving properties in the parabolic to elliptic limit) and simulations.

In a second part I will present the derivation of a new discrete Trudinger inequality applicable in this setting, which allows to show that for small mass, the discrete version of the Keller-Segel entropy remains bounded from below for solutions of the scheme, independently of the mesh size.

This is based on joint works with Ariane Trescases and Antoine Zurek.

12:00 Lunch

6 Charles Collot: Determination of the long-time behaviour of solutions to the two-dimensional Keller-Segel system with critical mass

This presentation will describe the dynamics of solutions to the parabolic-elliptic Keller-Segel system in two dimensions. A first behaviour is that the solution is global and converges to a forward self-similar solution, which occurs if the mass of the solution is less than the explicit threshold of 8 pi [Blanchet-Dolbeault-Perthame and related works]. A second behaviour is that the solution blows up in finite time, which happens if the mass is more than 8 pi. Explicit examples are given by the collapse of a single stationary state [Herrero-Velazquez] or the collision of several collapsing stationary states [Collot-Ghoul-Masmoudi-Nguyen].

The presentation will focus on the case of the critical mass of 8 pi, for solutions with finite second moment. Such solutions are global and concentrate in infinite time [Carrillo-Masmoudi], and an explicit example of a solution concentrating a stationary state at a scale log(t)^(-1/2) was constructed in the radial class [Ghoul-Masmoudi] and in the non-radial class [Davilá-del Pino-Dolbeault-Musso-Wei]. The main result that will be presented shows that this is the universal dynamics at critical mass, i.e. that all general solutions will converge to a stationary state which concentrates at scale log(t)^(-1/2) around the center of mass of the solution. The proof combines soliton resolution techniques, with the control of the evolution of the solution in various critical spaces, and with the perturbative analysis around an approximate solution involving multiple scales. This is based on joint work with Federico Buseghin 1

7 Philippe Souplet: Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

We study the space-time concentration asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel system of chemotaxis, in a ball or in the whole space. We show that, in dimensions $3\le n\le 9$, for any solution $u$ blowing at time $T$, there exists a nonflat backward self-similar solution U such that $u/U\to 1$ as $(x,t)\to(0,T)$.
This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables $(x,t)$, which was previously known only in the microscopic scale $|x| \le O(\sqrt{T-t})$. As a consequence, we obtain the sharp final profile: $\lim |x|^2 u(x,T) = L > 0$ as $x\to 0$, as well as the precise two-sided space-time global estimates.

15:30 Coffee & Discussions