Acoustic limit for the Boltzmann equation in the whole space

This paper is devoted to the following rescaled Boltzmann equation in the acoustic time scaling in the whole spaceequation(0.1)∂tFϵ+v⋅∇xFϵ=1ϵQ(Fϵ,Fϵ),x∈R3,t>0, with prescribed initial dataFϵ|t=0=Fϵ(0,x,v),x∈R3. For a solutionFϵ(t,x,v)=μ+μϵfϵ(t,x,v) to the rescaled Boltzmann equation (0.1) in the whole space R3R3 for all t⩾0t⩾0 with initial dataFϵ(0,x,v)=F0ϵ(x,v)=μ+μϵfϵ(0,x,v),x,v∈R3, our main purpose is to justify the global-in-time uniform energy estimates of fϵ(t,x,v)fϵ(t,x,v) in ϵ   and prove that fϵ(t,x,v)fϵ(t,x,v) converges strongly to f(t,x,v)f(t,x,v) whose dynamic is governed by the acoustic system.