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{lunate} + v ṡ ∇_x F^ε{lunate} = frac(1, ε{lunate}) Q (F^ε{lunate}, F^ε{lunate}), x ∈ R³, t > 0, with prescribed initial dataF^ε{lunate} |_{t = 0} = F^ε{lunate} (0, x, v), x ∈ R³ . For a solutionF^ε{lunate} (t, x, v) = μ + sqrt(μ) ε{lunate} f^ε{lunate} (t, x, v) to the rescaled Boltzmann equation (0.1) in the whole space R³ for all t ≥ 0 with initial dataF^ε{lunate} (0, x, v) = F₀^ε{lunate} (x, v) = μ + sqrt(μ) ε{lunate} f^ε{lunate} (0, x, v), x, v ∈ R³, our main purpose is to justify the global-in-time uniform energy estimates of f^ε{lunate} (t, x, v) in ε{lunate} and prove that f^ε{lunate} (t, x, v) converges strongly to f (t, x, v) whose dynamic is governed by the acoustic system. © 2009 Elsevier Inc. All rights reserved.