Julien Barré
Vlasov equation close to marginal stability: Finite-size fluctuations
ABSTRACT: Kinetic equations of Vlasov type are usually obtained as large size limits of Hamiltonian systems of particles. Finite size fluctuations around the deterministic Vlasov limit are a natural question of interest and an important ingredient in kinetic theories. Close to marginal stability, Central Limit Theorem breaks down. What should replace it? I will present present some studies on this open problem, mainly through numerical simulations and heuristic computations.
It is a joint work with Yoshiyuki Yamaguchi (Kyoto University, Japan).
Léo Bigorgne
Modified scattering for the Vlasov-Poisson system with initial and scattering data in the same space
ABSTRACT: In this talk, we will first review recent results on the long-time dynamics of the Vlasov–Poisson system. Then, we will see how to prove modified scattering for small data solutions in a functional framework where the initial data, scattering states, and asymptotic convergence are measured in the same topology.
Emanuele Caglioti
Random matching for densities with unbounded support: some recent results
ABSTRACT: I will consider the Euclidean Random Matching problem for densities with unbounded support.
In a challenging paper, Caracciolo et. al., on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is logN / 2πN plus corrections.This and other related conjectures has been proved by Ambrosio et al. in a series of challenging papers.
In the talk I will review the results cited above and then I will focus on the case of radial densities on unbounded domains, e.g. the Gaussian, in generic dimension and exponent p.
Anwar El Rhirhayi
Capturing Dynamical Fluctuations in Long-Range Interacting Systems: A Stochastic Differential Equations Approach
ABSTRACT: The (deterministic) Landau equation captures the mean long-term evolution of long-range weakly interacting finite-$N$ systems.
Though successful, this kinetic equation fundamentally ignores dynamical fluctuations.
Building upon Large Deviation theory, we present a physically consistent system of stochastic differential equations that simultaneously recovers the mean Landau dynamics, but also accurately captures the corresponding fluctuations among different realisations.
We show in particular how these stochastic equations arise as a diffusive limit of many weak two-body deflections.
We extensively validate these equations against tailored direct $N$-body simulations, showing an exquisite level of agreement.
Megan Griffin-Pickering
A probabilistic mean-field derivation for the ionic Vlasov-Poisson system
ABSTRACT: The ionic Vlasov-Poisson system describes ions in a dilute plasma interacting with a thermalized electron background. Compared to the more well-known electron Vlasov-Poisson system, the ionic model includes an additional exponential nonlinearity in the Poisson coupling for the electrostatic potential, which creates new mathematical challenges.
While the system can be formally derived through a mean-field limit from a microscopic system of ions, it is an open problem to justify this limit rigorously. Previous results on the derivation of the 3D ionic Vlasov--Poisson system required a truncation of the singularity in the Coulomb interaction at spatial scales of order $N^{- \beta}$ with $\beta < 1/15$. Meanwhile, the electron model has been successfully derived under less restrictive truncation conditions, e.g. $\beta < 1/3$ (Lazarovici-Pickl, ARMA 2017) or $\beta < 5/12$ (Feistl-Held-Pickl, arXiv:2504.01471), using `probabilistic' methods.
I will discuss recent results in which the Vlasov-Poisson system for ions is derived from a microscopic system of ions and thermalized electrons with interaction truncated at scale $N^{- \beta}$ with $\beta < 1/3$. The proof involves a generalisation of the probabilistic approach to mean-field limits to systems with interaction forces defined through an implicit, nonlinear coupling, by developing quantified law of large numbers estimates for uniform convergence of convolutions with empirical measures.
Grace Mattingly
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ABSTRACT:
Mario Pulvirenti
Is the BGK a toy model?
ABSTRACT: In this talk I consider the well known BGK equation, describing a rarefied gas close to the hydrodynamic regime. The starting point is the stochastic process associated to a particle system, which is known to be close to the mechanical behavior in a low-density regime. We (this is a work in collaboration with P. Butta’ and S. Simonella) plan to derive the BGK equation under a suitable scaling, giving a precise physical meaning to such an equation.
Stefano Rossi
Stability of vacuum in the screened Vlasov-Poisson equation
ABSTRACT: I will describe the long-time behavior of small-data dispersive solutions to the screened Vlasov--Poisson equation. In contrast to the unscreened case, shielding effects yield improved decay properties for the electric field, making it possible to study the system in lower dimensions. I will focus on recent results regarding the asymptotic behavior of solutions in one and two dimensions.
Frédéric Rousset
Semiclassical limit of the cubic Nonlinear Schrodinger Equation for mixed states
ABSTRACT: We shall present a recent work on the semiclassical limit of NLS and Hartree type equations for infinitely many particles under suitable assumptions on the Wigner transform of the initial datum. The formal limit is a singular kinetic equation of Vlasov type where the force field is given by the gradient of the density. We will recall previous results on the well-posedness theory of this type of kinetic equations and explain how to get uniform estimates suitable to justify the semiclassical limit. Joint work with Daniel Han-Kwan (Nantes).
Elena Salguero
Continuation Criteria for the Relativistic Vlasov–Maxwell System
ABSTRACT: The relativistic Vlasov–Maxwell (rVM) system is a fundamental model in kinetic theory describing the dynamics of collisionless plasmas with high-velocity particles. In the 1980s, Glassey and Strauss proved that solutions remain regular as long as the momentum support remains bounded in time, providing a key continuation criterion. Understanding the evolution of particle momentum is therefore central to establishing global existence of solutions. In this talk, we review known continuation criteria and present new insights in this direction. This is based on joint work with M. Hernandez and N. Patel.
Renato Velozo Ruiz
Phase mixing and the Vlasov equation in cosmology
ABSTRACT: In this talk, we will consider the Vlasov equation on expanding isotropic homogeneous tori described by the Friedmann–Lemaitre–Robertson–Walker cosmological spacetimes. For expansion rate $t^q$, with $0 < q \leq 1/2$ (excluding some exceptional values), we show that the difference of the spatial density with the spatial average decays with a quantitative rate depending on the regularity of the initial datum. This is joint work with Martin Taylor.