The hydrodynamical limit of the non linear Boltzmann equation

This paper is devoted to the connection between kinetic theory and macroscopic fluid dynamics. Formal limits are systematically derived and a rigorous results are given concerning the validity of these limits. The Broadwell model and the Cabannes model of the Boltzmann equation for a simple discrete velocity gas are investigated in the Hydrodynamical limit to the level of the heat equation locally in time as the mean free path∊ → 0. The nonlinear Kac equation for a rarefied gas is investigated in the fluid dynamical limit to the level of the linear equation locally in time, as the Knudsen number ∊ tends to zero.