From Boltzmann’s kinetic theory to Euler’s equations

Abstract

The incompressible Euler equations are obtained as a weak asymptotics of the Boltzmann equation in the fast relaxation limit (the Knudsen number Kn goes to zero), when both the Mach number Ma (defined as the ratio between the bulk velocity and the speed of sound) and the inverse Reynolds number Kn / Ma (which measures the viscosity of the fluid) go to zero. The entropy method used here consists in deriving some stability inequality which allows us to compare the sequence of solutions of the scaled Boltzmann equation to its expected limit (provided that it is sufficiently smooth). It thus leads to some strong convergence result. One of the main points to be understood is how to deal with the corrections to the weak limit, i.e. the contributions converging weakly but not strongly to 0 such as the initial layer or the acoustic waves.