There is a PhD or postdoctoral position available at Centrale Supélec Université Paris-Saclay.

Topic description: Chemotaxis describes the directed motion of cells toward higher concentrations of a chemical signal. The Keller–Segel model is a canonical framework capturing this mechanism and has a rich mathematical theory, notably in two dimensions where critical mass phenomena and concentration can occur. In many realistic environments, however, the motion takes place in a bounded region and cells may leave or die upon reaching the boundary. This motivates the study of Keller–Segel dynamics with absorbing boundary conditions, where the total mass is not conserved and boundary effects influence both the interaction structure and the long-time behavior.

From a mathematical viewpoint, this setting raises intertwined questions at the interface of interacting particle systems, singular stochastic dynamics, nonlinear PDEs, and functional inequalities: How does absorption modify criticality? Can microscopic absorbed particle models converge to macroscopic absorbed PDEs? Which stability and uniqueness mechanisms ensure deterministic mean-field limits?

In two dimensions, the parabolic–elliptic Keller–Segel model exhibits a delicate balance between diffusion and attraction, leading to regimes of global existence as well as concentration and blow-up for large initial mass under mass-preserving settings. A major body of work characterizes blow-up points, minimal concentration mass, and entropy-based criteria, often relying on sharp functional inequalities and localized compactness arguments. On bounded domains, the chemoattractant typically solves an elliptic boundary value problem, which induces boundary-dependent interaction forces through the associated Green function. Absorbing (Dirichlet) boundary conditions for the density naturally enforce strict mass loss over time. Compared with the whole-space model, this modifies the mechanism and possibly the thresholds of concentration.

This PhD topic concerns a boundary-absorbed variant of the two-dimensional Keller–Segel chemotaxis model in bounded domains. The project investigates the link between (i) microscopic interacting particle systems with singular attraction and absorption (killing) at the boundary, and (ii) macroscopic nonlinear PDE limits with Dirichlet boundary conditions and strict mass loss. A central focus is the role of absorption in the classical criticality/blow-up mechanisms of Keller–Segel, and the development of a unified probabilistic–PDE framework providing well-posedness, stability, and rigorous particle-to-PDE convergence results, while keeping the modelling assumptions compatible with applications.

Candidate profile: Strong background in applied mathematics, with interest in probability and/or PDEs. Useful topics include: stochastic differential equations, weak convergence/tightness, interacting particle systems, parabolic and elliptic PDEs, and functional inequalities. Prior exposure to Keller–Segel or mean-field limits is a plus. The candidate is suggested to start with a «mémoire» before starting the thesis.

Contact: This thesis is supposed to start no later than 31 December 2026, and is of duration 3 years, the postdoc is for 2 years (funded by ANR JCJC). Please email a CV and a short statement of interest (and optionally transcript + reference contacts) to Gaoyue Guo — gaoyue.guo@centralesupelec.fr