H.B. Keller Colloquium

We will give a short introduction to the motivation and problems in kinetic theory, in particular mean-field limits and hydrodynamic limits. This touches on the fundamental problem of giving a rigorous justification of the main PDE in physics. There are many interesting mathematical ideas involved, and in particular we will talk about hypocoercivity techniques, used when studying convergence to equilibrium in many kinetic equations. For a more concrete research problem, we will consider linear kinetic equations of the form $\partial_t f + \frac{1}{\epsilon} v \nabla_x f = \frac{1}{\epsilon^2} L(f)$, for an unknown $f$ which depends on time $t$, position $x$ and velocity $v$, and where $L$ is a linear operator which acts only in the velocity variable, and which typically has a probability equilibrium in $v$. Important examples include the Fokker-Planck operator, nonlocal diffusion operators, linear BGK-type operators, or linear Boltzmann operators. This PDE typically represents a mesoscopic physical model, where we keep track of the probability distribution of the position and velocity of particles. It is well known that when $\epsilon$ tends to $0$, this type of equation has a macroscopic or diffusive limit for the density $\rho(t,x) := \int f(t,x,v) dv$, which is either the standard heat equation, or the fractional heat equation. As a new result, we show that for a fixed epsilon, the behavior of this equation for large times also follows the standard or fractional heat equation, and that the long-time and small-epsilon limits are actually interchangeable in many cases. This is a work in collaboration with Stéphane Mischler (U. Paris-Dauphine) and Niccolò Tassi (U. Granada).