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 spaceequation(0.1)ϵ∂tFϵ+v⋅∇xFϵ=1ϵQ(Fϵ,Fϵ),x∈RN,v∈RN,t>0, with prescribed initial dataequation(0.2)Fϵ(t,x,v)|t=0=F0ϵ(x,v),x∈RN,v∈RN. Our main purpose is to justify the global validity of the diffusive expansionequation(0.3)Fϵ(t,x,v)=μ+μ{ϵf1(t,x,v)+ϵ2f2(t,x,v)+⋯+ϵn−1fn−1(t,x,v)+ϵnfnϵ(t,x,v)} for a solution Fϵ(t,x,v)Fϵ(t,x,v) of the rescaled Boltzmann equation (0.1) in the whole space RNRN for all t⩾0t⩾0 with initial data F0ϵ(x,v) satisfying the initial expansionequation(0.4)F0ϵ(x,v)=μ+μ{ϵf1(0,x,v)+ϵ2f2(0,x,v)+⋯+ϵn−1fn−1(0,x,v)+ϵnfnϵ(0,x,v)},(x,v)∈R2N. Here μ(v)=(2π)−N2exp(−|v|22) is a normalized global Maxwellian.Under the assumption that the fluid components of the coefficients fm(0,x,v)fm(0,x,v)(1⩽m⩽n)(1⩽m⩽n) of the initial expansion F0ϵ(x,v) have divergence-free velocity fields um0(x) as well as temperature fields θm0(x), if we assume further that the velocity-temperature fields [u10(x),θ10(x)] of f1(0,x,v)f1(0,x,v) have small amplitude in Hs(RN)Hs(RN)(s⩾2(N+n+2))(s⩾2(N+n+2)), we can determine these coefficients fm(t,x,v)fm(t,x,v)(1⩽m⩽n)(1⩽m⩽n) in the diffusive expansion (0.3) uniquely by an iteration method and energy method. The hydrodynamic component of these coefficients fm(t,x,v)fm(t,x,v)(1⩽m⩽n)(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 RNRN and this difficulty is overcome by using the Lp–LqLp–Lq-estimate on the Riesz potential. Moreover, by exploiting the energy method, we can also deduce certain the space–time energy estimates on these coefficients fm(t,x,v)fm(t,x,v)(1⩽m⩽n)(1⩽m⩽n).Once the coefficients fm(t,x,v)fm(t,x,v)(1⩽m⩽n)(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 ϵ   on the remainders fnϵ(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 thatequation(0.5)N>2n+2.N>2n+2. Notice that for m⩾2m⩾2, um(t,x)um(t,x) is no longer a divergence-free vector and it is worth to pointing out that, for m⩾3m⩾3, it was in deducing certain estimates on pm(t,x)pm(t,x) by the Lp–LqLp–Lq-estimate on the Riesz potential that we need to require that N>2n+2N>2n+2.