arXiv subjects

Pierre-Emmanuel Jabin

Publications and source records attributed to Pierre-Emmanuel Jabin.

At least 19 recordsLinked to original sources

arXiv2026

Derivation of 2D Vlasov-Poisson for classical particles with Coulomb interactions

We obtain the first derivation of the 2D Vlasov-Poisson equation for classical particles with Coulomb interactions without any microscopic cutoff. The proof relies on our recent dual hierarchical approach to mean-field limits, together with a refined analysis of dual BBGKY hierarchies on linearized correlations, based on kinetic regularization effects. The result holds more generally for arbitrary singular interaction forces $K\in L^{2-\eta}_{loc}$ with $\eta>0$ small enough, in any dimension, and it extends to Brownian particles. It holds globally in time as long as the mean-field solution is regular enough.

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arXiv2026

Singular mean-field limits for fluctuations around equilibrium

This work addresses the mean-field limit of inertial particle systems with singular interactions in a perturbative regime around Gibbs equilibrium. We prove that small fluctuations around equilibrium are asymptotically governed by the linearized Vlasov equation. The result applies to a broad class of singular interaction kernels, including the Coulomb case in dimensions $d\le3$. In particular, this provides a rigorous derivation of the linearized mean-field dynamics near equilibrium in settings where the corresponding nonlinear mean-field limit remains out of reach.

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arXiv2026

Quantitative Estimates for Mean-Field Limits and Correlation Functions through a Duality Framework

We investigate the mean-field limit for interacting particle systems through a duality-based framework and obtain quantitative estimates on the convergence of marginals as well as on correlation functions. The analysis applies to second-order Vlasov systems with pairwise interactions under a mean-field scaling and relies on a hierarchy of dual cumulants associated with the particle dynamics. In particular, for merely square--integrable interaction forces, we derive the natural fluctuation--scale rate $\mathcal{O}(N^{-1/2})$. By introducing an iterative argument on the hierarchy of dual cumulants, we leverage this bound to recover the optimal mean-field rate $\mathcal{O}(N^{-1})$ and to obtain robust estimates on the dual cumulants, at the expense of corresponding regularity assumptions on the interaction kernel. Finally, using the relation between dual and direct correlations, we transfer these bounds to direct cumulants, yielding refined information on correlations and deviations from chaos. The approach provides a unified framework for simultaneously controlling the mean-field limit and the higher-order correlation structure of the particle system.

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arXiv2025

Correlation estimates for Brownian particles with singular interactions

We study particle systems with singular pairwise interactions and non-vanishing diffusion in the mean-field scaling. A classical approach to describing corrections to mean-field behavior is through the analysis of correlation functions. For bounded interactions, the optimal estimates on correlations are well known: the $m$-particle correlation function is $G_{N,m}=O(N^{1-m})$ for all $m$. Such estimates, however, have remained out of reach for more singular interactions. In this work, we develop a new framework based on linearized correlation functions, which allows us to derive robust bounds for systems with merely square-integrable interaction kernels, providing the first systematic control of correlations in the singular setting. Although at first not optimal, our estimates can be partially refined a posteriori using the BBGKY hierarchy: in the case of bounded interactions, our method recovers the known optimal estimates with a simplified argument. As key applications, we establish the validity of the Bogolyubov correction to mean field and prove a central limit theorem for the empirical measure, extending these results beyond the bounded interaction regime for the first time.

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arXiv2025

Multiscale Analysis of a Kinetic Model of Confined Suspensions of Self-Propelled Rods

The behavior of active matter under confinement poses significant challenges due to the intricate coupling between dynamics near boundaries and those in the bulk. A defining feature of active matter systems is that a substantial portion of their dynamics takes place near confining boundaries. In our previous work, we developed a kinetic framework that enables direct computation of the probability distribution functions for both the position and orientation of active rods. A distinguishing aspect of this approach is its explicit treatment of wall accumulation through the use of two coupled probability distribution functions: one describing the bulk population and the other representing rods accumulated at the boundary. Another novel feature is the structure of the governing equation, which is degenerate: it is second-order in one non-temporal variable and first-order in another. The main focus of this paper is to rigorously justify this model via multi-scale analysis. We first establish well-posedness of the system and then employ two distinct multi-scale derivations to obtain the model as a singular limit of a more classical kinetic system in the regime of vanishing translational diffusion. For analytical clarity, we consider the case in which active rods, once accumulated at the wall, remain permanently confined there. This work provides a rigorous mathematical foundation for reduced kinetic models of confined active matter, bridging microscopic dynamics and macroscopic accumulation phenomena.

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arXiv2025

Singular flows with time-varying weights

We study the mean field limit for singular dynamics with time evolving weights. Our results are an extension of the work of Serfaty \cite{duerinckx2020mean} and Bresch-Jabin-Wang \cite{bresch2019modulated}, which consider singular Coulomb flows with weights which are constant time. The inclusion of time dependent weights necessitates the commutator estimates of \cite{duerinckx2020mean,bresch2019modulated}, as well as a new functional inequality. The well-posedness of the mean field PDE and the associated system of trajectories is also proved.

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arXiv2025

A duality method for mean-field limits with singular interactions

We introduce a new approach to derive mean-field limits for first- and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at possibly vanishing temperature. In case of first-order systems, it allows to recover in particular the mean-field limit to the 2d Euler and Navier-Stokes equations. The approach also provides convergence rates.

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arXiv2024

Longtime and chaotic dynamics in microscopic systems with singular interactions

This paper investigates the long time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of $N$ interacting point particles, where the time evolution of the joint distribution $f_N(t)$ is governed by the Liouville equation. Our primary objective is to analyze the system's behavior over extended time intervals, focusing on stability, potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime where $N \gg 1$, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases it is possible to derive uniform bounds in $ L^2 $, both over time and with respect to the number of particles, on the marginals $ \left(f_{k,N}\right)_{1\leq k \leq N}$, irrespective of the initial state's chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels surpassing the traditional $L^d$ regularity barriers, $K \in W^{\frac{-2}{d+2},d+2}(\mathbb{T}^d)$, where $\mathbb{T}$ denotes the $1$-torus and $d\geq2$ is the dimension. Finally, we address the highly singular case of $K \in H^{-1}(\mathbb{T}^d)$ within high-temperature regimes, offering new insights into the behavior of such systems.

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arXiv2024

Fitting parameters of a Fokker-Planck-like equation with constraint

We analyse a Fokker-Planck like equation, driven by a scalar parameter in order to reach an integral constraint. We exhibit criteria guaranteeing existence-uniqueness of a solution. We also provide counter-examples. This problem is motivated by an application to the immune control of tumor growth.

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arXiv2024

Relative Entropy Method for Particle Approximation of the Landau Equation for Maxwellian Molecules

We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.

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arXiv2024

The graph limit for a pairwise competition model

This paper is aimed at extending the graph limit with time dependent weights obtained in [1] for the case of a pairwise competition model introduced in [10], in which the equation governing the weights involves a weak singularity at the origin. Well posedness for the graph limit equation associated with the ODE system of the pairwise competition model is also proved.

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arXiv2024

Dense networks of integrate-and-fire neurons: Spatially-extended mean-field limit of the empirical measure

The dynamics of spatially-structured networks of $N$ interacting stochastic neurons can be described by deterministic population equations in the mean-field limit. While this is known, a general question has remained unanswered: does synaptic weight scaling suffice, by itself, to guarantee the convergence of network dynamics to a deterministic population equation, even when networks are not assumed to be homogeneous or spatially structured? In this work, we consider networks of stochastic integrate-and-fire neurons with arbitrary synaptic weights satisfying a $O(1/N)$ scaling condition. Borrowing results from the theory of dense graph limits, or graphons, we prove that, as $N\to\infty$, and up to the extraction of a subsequence, the empirical measure of the neurons' membrane potentials converges to the solution of a spatially-extended mean-field partial differential equation (PDE). Our proof requires analytical techniques that go beyond standard propagation of chaos methods. In particular, we introduce a weak metric that depends on the dense graph limit kernel and we show how the weak convergence of the initial data can be obtained by propagating the regularity of the limit kernel along the dual-backward equation associated with the spatially-extended mean-field PDE. Overall, this result invites us to reinterpret spatially-extended population equations as universal mean-field limits of networks of neurons with $O(1/N)$ synaptic weight scaling.

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arXiv2024

Mean-field derivation of Landau-like equations

We derive a class of space homogeneous Landau-like equations from stochastic interacting particles. Through the use of relative entropy, we obtain quantitative bounds on the distance between the solution of the N-particle Liouville equation and the tensorised solution of the limiting Landau-like equation.

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arXiv2024

Dynamics of point-vortex type systems near thermal equilibrium: relaxation or not?

This article is devoted to the long-time dynamics of point-vortex type systems near thermal equilibrium and to the possible emergence of collisional relaxation. More precisely, we consider a tagged particle coupled to a large number of background particles that are initially at equilibrium, and we analyze its resulting slow dynamics. On the one hand, in the spirit of the Lenard-Balescu relaxation for plasmas, we establish in a generic setting the outset of the slow thermalization of the tagged particle. On the other hand, we show that a completely different phenomenology is also possible in some degenerate regime: the slow dynamics of the tagged particle then remains conservative and the thermalization no longer holds in a strict sense. We provide the first detailed description of this degenerate regime and of its mixing properties. Note that it is particularly delicate to handle due to statistical closure problems, which manifest themselves as a lack of self-adjointness of the effective Hamiltonian.

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arXiv2023

The mean-field Limit of sparse networks of integrate and fire neurons

We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the 1-particle distribution, for a large number of neurons with no structural assumptions on the connection map outside of a generalized mean-field scaling. We propose a novel notion of observables that naturally extends the notion of marginals to systems with non-identical or non-exchangeable agents. Our new observables satisfy a complex approximate hierarchy, essentially a tree-indexed extension of the classical BBGKY hierarchy. We are able to pass to the limit in this hierarchy as the number of neurons increases through novel quantitative stability estimates in some adapted weak norm. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution.

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arXiv2023

Hölder and Sobolev regularity of optimal transportation potentials with rough measures

We consider a Kantorovich potential associated to an optimal transportation problem between measures that are not necessarily absolutely continuous with respect to the Lebesgue measure, but are comparable to the Lebesgue measure when restricted to balls with radius greater than some $δ>0$. Our main results extend the classical regularity theory of optimal transportation to this framework. In particular, we establish both Hölder and Sobolev regularity results for Kantorovich potentials up to some critical length scale depending on $δ$. Our assumptions are very natural in the context of the numerical computation of optimal maps, which often involves approximating by sums of Dirac masses some measures that are absolutely continuous with densities bounded away from zero and infinity on their supports.

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arXiv2023

Global Existence of Weak Solutions for Compresssible Navier--Stokes--Fourier Equations with the Truncated Virial Pressure Law

This paper concerns the existence of global weak solutions {\it à la Leray} for compressible Navier--Stokes--Fourier systems with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

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arXiv2022

Discretizing advection equations with rough velocity fields on non-cartesian grids

We investigate the properties of discretizations of advection equations on non-cartesian grids and graphs in general. Advection equations discretized on non-cartesian grids have remained a long-standing challenge as the structure of the grid can lead to strong oscillations in the solution, even for otherwise constant velocity fields. We introduce a new method to track oscillations of the solution for rough velocity fields on any graph. The method in particular highlights some inherent structural conditions on the mesh for propagating regularity on solutions.

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