Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas

Abstract

We consider the kinetic transport equation that arise in the Boltzmann-Grad limit of the two-dimensional periodic Lorentz Gas. This equation has been obtained by extending the phase space of positions and velocities through the introduction of two new variables, representing respectively the time to the next collision and the corresponding impact parameter. Here we mostly focus on the case of periodic boundary conditions on the positions space: we prove that, under suitable hypothesis, the time evolution of a probability density on the extended phase space converges to the equilibrium state with respect to the $L^p$ norm ($^*$-weakly if $p=\infty$), if such initial density is $L^p$. If $p=2$, or if the initial datum does not depend on the position, we also get more precise estimates about the rate of the approach to the equilibrium. Our proof is based on the analysis of the long time behavior of the Fourier coefficients of the solution.