Boltzmann diffusive limit beyond the Navier-Stokes approximation

Given a normalized Maxwellian mu and n >= 1, we establish the global-in-time validity of a diffusive expansion (0.1) F-epsilon(t, x, v) = mu + root mu {epsilon f(1)(t, x, v) + epsilon(2) f(2)(t, x, v) +(...)+ epsilon(n) f(n)(epsilon)(t, x, v)}, for a solution F-epsilon to the resealed Boltzmann equation (diffusive scaling) (0.2) epsilon partial derivative F-t(epsilon) + v (.) del F-x(epsilon) = 1/epsilon Q(F-epsilon, F-epsilon) inside a periodic box T-3. We assume that in the initial expansion (0.1) at t = 0, the fluid parts of these f(m) (0, x, v) have arbitrary divergence-free velocity fields as well as temperature fields for all 1 < m < n while f(1) (0, x, v) has small amplitude in H-2. For m >= 2, these f(m) (t, x, v) are determined by a sequence of linear Navier-Stokes-Fourier systems iteratively. More importantly, the remainder f(n)(epsilon) (t, x, v) is proven to decay in time uniformly in epsilon via a unified nonlinear energy method. In particular, our results lead to an error estimate for f(1) (t, x, v), the well-known Navier-Stokes-Fourier approximation, and beyond. The collision kernel Q includes hard-sphere, the cutoff inverse-power, as well as the Coulomb interactions. (c) 2005 Wiley Periodicals, Inc.