Abstract: We use some new nonlinear estimates found in [LM] to improve the results of [GL] that establish the acoustic limit for DiPerna-Lions solutions of Boltzmann equation in three ways. First, we enlarge the class of collision kernels treated to that found in [LM], thereby treating all classical collision kernels to which the DiPerna–Lions theory applies. Second, we improve the scaling of the kinetic density fluctuations with Knudsen number from O(ε^m) for some m>½ to O(ε^½). Third, we extend the results from periodic domains to bounded domains with impermeable boundaries, deriving the boundary condition for the acoustic system.

[LM] C.D. Levermore and N. Masmoudi, From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System, Submitted to Arch. Ration. Mech. & Anal., 2008.
[GL] F. Golse and C.D. Levermore, The Stokes–Fourier and Acoustic Limits for the Boltzmann Equation, Commun. on Pure & Appl. Math. 55 (2002), 336–393.