Publications are listed by thematic. But they are also sorted by anti-chronological order on that page.

Many Particle System and Mean Field Limits.

• Propagation of chaos for the Landau equation with moderately soft potentials, with Nicolas Fournier, 60 p. (arXiv)
  We give two proof one propagation of molecular chaos for the Landau equation for moderately soft potentials. The first one provide a rate of convergence, and is valid when the gamma (with the usual notation) is between -1 and 0. It relies on a weak strong stability estimates for the Landau equation, that we are able to apply (in some sense) to the particle system. The second one is an adaptation of our work with Sthéphane Mischler on the 2D Navier-stokes equation and provide qualititative propagation of chaos when gamma lies between -2 and 0. The main difficulty is that the dissipation is not uniform in the LAndau equation, and we have to introduce a perturbed particle system (fully dissipative) to conclude.
• The derivation of Swarming models: Mean-Field Limit and Wasserstein distances, with Jose Antonio Carrillo and Young-Pil Choi, in a CISM Volume 553, 2014, pp 1-46 (Springer) (DOI, HAL, arXiv)
  We used the methods develop with PE Jabin, in the setting of aggregation equation. The intersting point is that the usual (for the Vlasov equation) uniform bounds are repalced by L^p bounds.
• Mean field limit for the one dimensional Vlasov-Poisson equation, in Séminaire Laurent Schwartz (2012-2013). (Séminaire's website, HAL, Arxiv)
  I proved a simple weak-strong stability result for the 1D VLasov-Poisson equation. It implies mean field limit and propagation of chaos. I introduce also (as others italians have already done) some notations borrowed for the SDE, that allow to give more simple proof (at least when you are used to).
• Propagation of chaos for the 2D viscous Vortex Model, with Nicolas Fournier and Stéphane Mischler, in J. Eur. Math. Soc. , Volume 16, Issue 7, 2014, pp. 1423–1466. (DOI, HAL, arXiv)
  We revisited a problem studied by H. Osada in the '80, and prove the propagation of molecular chaos for 2D system of vortices, with only small assumptions on the initial configuration. The proof rely heavily on the properties of the Fisher information, and in particular on a property proved in the next article.
• On Kac's chaos and related problems, with Stéphane Mischler, in J. Funct. Anal., Volume 266, Issue 10, (2014), Pages 6055–6157. (DOI, HAL, arXiv)
  A toolbox for Many particle Systems. Providing quantitative version of the equivalence between the different formulations of the propagation of (molecular) chaos. Studying important properties of the Entropy and Fisher information functionals, in the limit of large system.
• Propagation of chaos for particles approximations of Vlasov equations with singular forces, with Pierre-Emmanuel Jabin, to appear in Ann. Sci. Éc. Norm. Supér, (HAL, Arxiv)
  This article is an amelioration of the last article in that section. We prove the mean-field limit for 3D particles system interacting via singular force : in |x|^-a, with a in (0,1) near the origin. The conditions on the initial configuration are not strong : there are in some sense "generic" and it allow to obtain the propagation of chaos in that case. Only careful estimates with the help of the infinite Monge-Kantorovich-Wasserstein distance.
• Stability of trajectories for N-particles dynamics with singular potential, with Julien Barré and Pierre-Emmanuel Jabin, in a topical issue on Long-Range Interacting Systems of J. Stat. Mech. (2010) (DOI, HAL, Arxiv).
  Here we study the case of 3D particle system with singular interaction in |x|^-a, with a in (1,2) near the origin. We study the stability of Gibbs equilibrium in a torus. It is difficult to explain precisely what we obtain : we roughly show that the trajectories are stable with respect to small perturbation of the initial position. It implies some propagation of chaos for some very particular initial distribution, approaching an uniform (in position) Maxwellian (in speed) profile. Not much, but a first step towards higher singularities.
• Wasserstein distances for vortices approximation of Euler-type equation, in Math. Models Methods Appl. Sci. 19 (2009), no. 8, 1357--1384. (DOI, HAL)
  I prove mean-field limits for Euler-like equation, missing the right case because I was unable to treat the right singularity. First time I introduced the infinite MKW distance, which helps to simplify the previous methods develop with PE Jabin.
• N particles approximation of the Vlasov equations with singular potential, with Pierre-Emmanuel Jabin, in Arch. Rach. Mech. Analysis 183 (2007), no. 3, 489--524. (DOI, HAL, Arxiv)
  We prove the mean-field limit for 3D particles system interacting via singular force : in |x|^-a, with a in (0,1) near the origin. There is a strong condition on the initial configuration that is very restrictive. So that the result does not allows to conclude to propagation of chaos. Mostly made of careful estimates with the help of discrete infinite norms.

Gyro-kinetic models and quasi-neutral limit for plasmas.

• Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation, with Daniel Han-Kwan, to appear in Comm. Math. Phys. (HAL, arXiv)
  We prove that in the quasineutral limit (when the Debye length goes to zero) of the 1D Vlasov-Poisson equation, the equilibria that satisfy the Penrose instability criteria (and some others technical conditions) are very unstable. We also prove that one humped symmetric equilibria are stable in the same limit, up to the filtration of the plasma oscillation (or Langmuir waves). We also study the BGK waves for the VP quasi-neutral equation.
• Well-posedness of a diffusive gyro-kinetic model. with Anne Nouri, in Ann IHP (C) Non Linear Analysis Vol. 28, 2011, p. 529-550 (DOI, HAL, Arxiv).
  We study a 2D gyro-kinetic quasi-neutral Vlasov equation used in plasma physics for the modelization of plasma in core of Tokamak. In fact that equation, is more or less a collection of coupled 2D Euler equations. We show, that despites the lack of regularity due to the quasi-neutrality equation, global solutions exists. We also get a short time uniqueness result.
• Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions, with Anne Nouri and Philippe Ghendrih, in Kinet. Relat. Models 2 (2009), no. 4, 707--725. (DOI, Arxiv)
  We study the limit of Vlasov equation in a fixed and constant magnetic field, when the Larmor radius goes to zero (the Debye length is kept constant). We provide a not new, but simple, result of convergence. We also show formally that upon some adiabatic assumptions on the electrons and ions response to electric forces, the quasi-neutral equation transform into a interesting equation where a double gyro-average appears. We provide also some uniqueness results in the stationary case.

ODE with weakly regular vector-fields.

• A new proof of the uniqueness of the flow for ordinary differential equations with BV vector fields, with Claude Le Bris, in AMPA Volume 190, Issue 1 (2011), Page 91. (DOI, HAL, Arxiv)
  We prove uniqueness of the Regular Lagragian Flow associated to $BV$ vector-field by a direct method : direct means that the result is not a consequence of an uniqueness results on the associated transport or continuity equations. It i an extension of the method introduced in the previous paper with PL Lions.
• Deux remarques sur les flots géneralisés d'équations différentielles ordinaires, with C. Le Bris et Pierre-Louis Lions, in Comptes rendus mathématiques 344, 2007, no. 12, 759-764. (DOI, HAL)
  We prove uniqueness of the Regular Lagragian Flow associated to $BV$ vector-field by a direct method : direct means that the result is not a consequence of an uniqueness results on the associated transport or continuity equations. The key idea is to compare the position at time t of the solution of the ODE starting form $x$, to the averaged position of the solutions starting in a neighboorhood of $x$.
• On Liouville transport equation with a force field in BV_loc, in Comm. Partial Differential Equations 29 (2004), no. 1-2, 207--217. (DOI, arXiv, HAL)
  I extend the famous result obtained by DiPerna and Lions about the existence and uniqueness of solutions to ODE with W^1,1 vector-field, to the particular case of system of particles in interaction, with a singularity at the origin. Basically, the main assumption is that the force is integrable or repulsive near the origin.
• On Two-dimensional Hamiltonian Transport Equations with L^p _loccoefficients, in Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 4, 625--644. (DOI, arXiv, HAL)
  I extend a previous result of L. Bouchut and L. Desvillettes on 2D transport equation (paper). There results was valid for continuous vector-field, and mine is valid for L² vector-field, with an additionnal condition on the local direction of the force-field. That condition was not really elegant, and it was again ameliorated by G. Alberti, and S. Bianchini, and G. Crippa (Paper). Their new condition is that $b$ should satisfy a kind of Sard Lemma.

Quantum decoherence by interaction with the environment

• Decoherence for a heavy particle interacting with a light one: new analysis and numerics, with Riccardo Adami and Claudia Negulescu. (HAL, arXiv)

We study a quantum model describing the interaction of a heavy particle with a light one. In the limit of small mass ratio, we obtain rigorously a simple "instataneous" collision operator, acting on the density matric of the heavy one. We study some of its property and use it to perform soem fast numerical simulation, which show on this toy model the apparition of decoherence.