We shall revisit the Boltzmann equation for rarefied non-linear particle dynamics, of conservative or dissipative nature, and on the stochastic N-particle model, introduced by M. Kac.
Related to this equation, we consider a a probabilistic dynamics from generalizations to N-particle model which includes multi-particle interactions. From basic symmetries and invariances for a general class of stochastic interactions, we show existence and uniqueness of states and recover the longtime dynamics and decay rates approaching stable laws characterized by self-similar rescaling, with finite or infinity energy initial data. We classify the moments integrability and see that broad tails (Pareto type) attractors are possible.

There is a large class of applications to these models including classical elastic or inelastic Maxwell type interactions with or without a thermostat, and social dynamics such as information percolation models, or wealth distributions models with Pareto tail formation.

This is work in collaboration with A. Bobylev, C. Cercignani and H. Tharkabhushanam.

The link between various viscoelastic fluids models and the symmetric matrix Riccati differential equations can be a new device that brings to unified and proper ways of numerical treatments for the viscoelastic models. In this talk, we describe a few steps toward efficient numerical schemes for complex fluids simulation. First, we construct stable finite element discretizations using Eulerian--Lagrangian methods based on the Riccati formulations of the viscoelastic models. Then we develop a new multilevel time-marching scheme; together with adaptive time-stepping schemes and time parallel schemes, we can build efficient methods for complex fluids simulation. Furthermore, we discuss two robust and efficient multilevel solvers for Stokes-type systems arising at each time step in the Eulerian--Lagrangian discretization.