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Yoan Tardy

Publications and source records attributed to Yoan Tardy.

4 recordsLinked to original sources
arXiv2026

Strong well-posedness of a singular SDE for signed Coulomb particles

We consider an SDE system for signed Coulomb particles moving in $\mathbb R^2$. Due to the singular Coulomb interaction force, collisions between particles of opposite sign will happen in finite time. Upon collision, the colliding particles are removed from the system. Our main results are the existence and uniqueness of strong global solutions and the characterization of all possible collisions. The challenge of the proofs is to deal with the singularity of the interactions. We overcome this by using scaling invariance of the process and by putting together several tools from [FT25] developed for the similar Keller--Segel particle system.

math.AP↗
arXiv2023

Weak convergence of the empirical measure for the Keller-Segel model in both subcritical and critical cases

We show the weak convergence, up to extraction of a subsequence, of the empirical measure for the Keller-Segel system of particles in both subcritical and critical cases, for general initial conditions. This particle system consists of $N$ planar Brownian motions interacting through a Coulombian attractive force, which is quite singular. In the subcritical case, a stronger result has been established by Bresch-Jabin-Wang \cite{bjw} at the price of two simplifications: the whole space $\rr^2$ is replaced by a torus and the initial condition is assumed to be regular. In the subcritical case, our proof is fairly straightforward: we use a {\it two particles} moment argument, which shows that particles do not aggregate in finite time, uniformly in the number of particles. The critical case requires more work.

math.PR↗
arXiv2023

Collisions of the supercritical Keller-Segel particle system

We study a particle system naturally associated to the $2$-dimensional Keller-Segel equation. It consists of $N$ Brownian particles in the plane, interacting through a binary attraction in $θ/(Nr)$, where $r$ stands for the distance between two particles. When the intensity $θ$ of this attraction is greater than $2$, this particle system explodes in finite time. We assume that $N>3θ$ and study in details what happens near explosion. There are two slightly different scenarios, depending on the values of $N$ and $θ$, here is one: at explosion, a cluster consisting of precisely $k_0$ particles emerges, for some deterministic $k_0\geq 7$ depending on $N$ and $θ$. Just before explosion, there are infinitely many $(k_0-1)$-ary collisions. There are also infinitely many $(k_0-2)$-ary collisions before each $(k_0-1)$-ary collision. And there are infinitely many binary collisions before each $(k_0-2)$-ary collision. Finally, collisions of subsets of $3,\dots,k_0-3$ particles never occur. The other scenario is similar except that there are no $(k_0-2)$-ary collisions.

math.PR↗