Pierre Gervais
Biography
Short Bio:
Université de Lille
Title: Hydrodynamic limit of elastic kinetic equations by a spectral approach
Abstract: Among the 23 problems listed by D. Hilbert during the International Congress of Mathematicians in 1900, the 6th one concerns the derivation of macroscopic descriptions of fluids from their microscopic descriptions. One possible strategy involves going through an intermediate level of description called mesoscopic, or kinetic, such as the Boltzmann or Landau models. This is referred to as the problem of hydrodynamic limits.
In the early 1990s, C. Bardos, F. Golse, and D. Levermore proved that one could formally derive the Navier-Stokes equations from kinetic equations conserving mass, velocity, and energy, and dissipating entropy, and the specific cases of the Boltzmann and Landau equations were gradually and independently addressed over the following three decades, despite their common structure.
The work on hydrodynamic limits is partly constrained by tools dating back to the early days of Boltzmann theory in the 1960s, allowing only for solutions satisfying a very restrictive integrability assumption, but also by results established using non-constructive arguments. In the case of Cauchy theories of kinetic equations, these restrictions have been lifted thanks to modern tools of "enlargement theory" and hypocoercivity methods developed from the 2000s onwards, notably by C. Mouhot, S. Mischler, and M. Gualdani.
In this talk, I present a collaboration with Bertrand Lods in which we have, on the one hand, considered the question of hydrodynamic limit for a kinetic equation **under generic assumptions** close to those of Bardos-Golse-Levermore, thus unifying previous results, and, on the other hand, modernized the necessary spectral study using the new theories of enlargement and hypocoercivity, thus providing the first fully quantitative results of hydrodynamic limits.