Mathematical tools for kinetic equations

Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, their range of application has been considerably extended. First introduced as a means to unify various perspectives on fluid mechanics, they are now used in plasma physics, semiconductor technology, astrophysics, biology.... They all are characterized by a density function that satisfies a Partial Differential Equation in the phase space.

This paper presents some of the simplest tools that have been devised to study more elaborate (coupled and nonlinear) problems. These tools are basic estimates for the linear first order kinetic-transport equation. Dispersive effects allow us to gain time decay, or space-time $L^p$ integrability, thanks to Strichartz-type inequalities. Moment lemmas gain better velocity integrability, and macroscopic controls transform them into space $L^p$integrability for velocity integrals.

These tools have been used to study several nonlinear problems. Among them we mention for example the Vlasov equations for mean field limits, the Boltzmann equation for collisional dilute flows, and the scattering equation with applications to cell motion (chemotaxis).

One of the early successes of kinetic theory has been to recover macroscopic equations from microscopic descriptions and thus to be able theoretically to compute transport coefficients. We also present several examples of the hydrodynamic limits, the diffusion limits and especially the recent derivation of the Navier-Stokes system from the Boltzmann equation, and the theory of strong field limits.