From Boltzmann's kinetic theory to Euler's equations

Laure Saint-Raymond, Laboratoire Jacques-Louis Lions, Paris 6 University

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 to compare the sequence of solutions to the scaled Boltzmann equation with its expected limit (provided it is sufficiently smooth), and thus leads to some strong convergence result.

One of the main points to be understood is how to take into account 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.