The fluid dynamical limits of the linearized Boltzmann equation.

Abstract

The old question concerning the mathematical formulation of the fluid dynamic limits of kinetic theory is examined by studying the solution of the Cauchy problem for two differently scaled linearized Boltzmann equations on periodic domain as the mean free path of the particles becomes small. Under minimal assumptions on the initial data, by using an a priori estimate, it is possible, in a Hilbert space functional frame, to prove the weak convergence of solutions toward a function that has the form of an infinitesimal maxwellian in the velocity variable. The velocity moments of this function are then proved to satisfy either the linearized Euler or the Stokes system of equations (depending on the chosen scaling), by passing to the limit in the conservation relations derived from the Boltzmann equation. A theorem injecting continuously the intersection of certain weak spaces into a normed one is proved. Together with properties of the Euler semigroup, this allows to show strong convergence of the first three moments of the distribution function toward the macroscopic quantities density, bulk velocity and temperature, solutions of the linearized Euler system. The Stokes case is treated somewhat differently, through the introduction of a result, proved by using the adjoint formulation for linear kinetic equations, that extends the averaging theory of Golse-Lions-Perthame-Sentis. The desired convergence for the divergence-free component of the second moment toward the macroscopic velocity is then shown.