My research focuses on the study of problems arising from the broad context of statistical mechanics. More precisely, I am interested in the analysis of partial differential equations in kinetic theory.
Kinetic equations describe the evolution of a N-particle system at a mesoscopic level, which is an intermediate viewpoint between the Newtonian dynamic of the microscopic particles and a macroscopic, hydrodynamical model.
Part of my work concerns the analysis of singular limits for Vlasov-type equations. In particular, I’m working on validating the quasineutral limit for the Vlasov-Poisson equation and the Vlasov Poisson equation for massless electrons. I’m also interested in the relationship between the quasineutral limit and the mean field limit, i.e. the derivation of mesoscopic equations of Vlasov type from the N particle system described by the Newton dynamic.
More recently, I started studying the geometric structure of stationary solutions for the 3-dimensional gravitational Vlasov-Poisson system in the spirit of the geometric interpretation of 2D Euler.
I am also interested in quantization of measures, which concerns the approximation in some optimal way of a diffuse measure with discrete ones. The problem of quantization of probability measures arises in a great variety of contexts, such as signal processing, pattern or speech recognition, economics, and much more. In particular, I worked on a dynamical approach to the quantization problem in space dimensions one and two, using gradient flows to recast the problem through (possibly degenerate) parabolic equations. This study also led me to the necessity of understanding parabolic equations of ultrafast diffusion type. The natural questions are related to the well-posedness and asymptotic behaviour to such degenerate evolution equations.
The common root of these topics is the general task of finding a rigorous justification of the description of a large number of identical objects -typically physical particles as gas moleculs or ions and electrons in a plasma- via approximated models that describe the behavior of the “generic object” of a physical system. In terms of mathematical tools, I make extensive use of PDEs techniques, optimal transport, probability, calculus of variations, and Riemannian geometry.
For more information about some of my research interests, here are some expository notes:
- A gradient flow perspective on the quantization problem. PDE Models for Multi-Agent Phenomena, Springer INdAM Ser, 28 (2018), pages 145–165.
- Recent developments on the well-posedness theory for Vlasov-type equations (with Megan Griffin-Pickering). To be included in the proceedings of the conference "Particle Systems and Partial Differential Equations”, editions VI, VII and VIII.
- Recent developments on quasineutral limits for Vlasov-type equations (with Megan Griffin-Pickering). Recent advances in kinetic equations and applications, Springer INdAM Series. To appear.