Diffusive expansion for solutions of the Boltzmann equation in the whole space

This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space, with prescribed initial data. Our main purpose is to justify the global validity of the diffusive expansion, for a solution, of the rescaled Boltzmann equation, in the whole space RN for all with initial data, satisfying the initial expansion, is a normalized global Maxwellian. Under the assumption that the fluid components of the coefficients f_m(0,x,v) (1≤m≤n) of the initial expansion, have divergence-free velocity fields u_m⁰(x) as well as temperature fields θ^m₀(x), if we assume further that the velocity-temperature fields [u₁⁰(x),θ₁0(x)] of f₁(0,x,v) have small amplitude in H^s(R^N) (s≤2(N+n+2)), we can determine these coefficients f_m(t,x,v) (1≤m≤n) in the diffusive expansion (0.3) uniquely by an iteration method and energy method. The hydrodynamic component of these coefficients f_m(t,x,v) (1≤m≤n) satisfies the incompressible condition, the Boussinesq relations and/or the Navier-Stokes-Fourier system respectively, while the microscopic component of these coefficients is determined by a recursive formula. Compared with the corresponding problem inside a periodic box studied in Y. Guo (2006) [18], the main difficulty here is due to the fact that Poincaré's inequality is not valid in the whole space R^N and this difficulty is overcome by using the L^p-L^q-estimate on the Riesz potential. Moreover, by exploiting the energy method, we can also deduce certain the space-time energy estimates on these coefficients f_m(t,x,v) (1≤m≤n). Once the coefficients f_m(t,x,v) (1≤m≤n) in the diffusive expansion (0.3) are uniquely determined and some delicate estimates have been obtained, the uniform estimates with respect to ∈ on the remainders fn∈(t,x,v) are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion (0.3) in the large provided that. (0.5)N>2n+2. Notice that for m≤2, u_m(t,x) is no longer a divergence-free vector and it is worth to pointing out that, for m≤3, it was in deducing certain estimates on p_m(t,x) by the L^p-L^q-estimate on the Riesz potential that we need to require that N>2n+2. © 2010 Elsevier Inc.