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 space(0.1) ∂ t F ϵ + v ⋅ ∇ x F ϵ = 1 ϵ Q ( F ϵ , F ϵ ) , x ∈ R 3 , t > 0 , with prescribed initial data F ϵ | t = 0 = F ϵ ( 0 , x , v ) , x ∈ R 3 . For a solution F ϵ ( t , x , v ) = μ + μ ϵ f ϵ ( t , x , v ) to the rescaled Boltzmann equation (0.1) in the whole space R 3 for all t ⩾ 0 with initial data F ϵ ( 0 , x , v ) = F 0 ϵ ( x , v ) = μ + μ ϵ f ϵ ( 0 , x , v ) , x , v ∈ R 3 , our main purpose is to justify the global-in-time uniform energy estimates of f ϵ ( t , x , v ) in ϵ and prove that f ϵ ( t , x , v ) converges strongly to f ( t , x , v ) whose dynamic is governed by the acoustic system.