## 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.