Lecture 1: Incompressible Navier-Stokes limit of the Boltzmann equation

We will demonstrate the (formal) asymptotic expansion of the Boltzmann equation for small Knudsen number that yeilds the incompressible Navier-Stokes set of equations in the conventional fluid dynamics [1]. The key is the so-called diffusive scaling. The derived continuity and momentum equations are indeed identical to those of the incompressible NS set and can be solved independently from the energy equation. However, the energy equation exhibits the compressibility and accordingly applying the incompressibility in the energy equation is not consistent. This resembles the feature in small Mach number expansion of the compressible NS set of equations in the viscous conservation law. The talk is based on Sone's monograph on Moleular Gas Dynamics [1].

[1] Y. Sone, Molecular Gas Dynamics (Birkhauser, Boston, 2007), Sec. 3.7.2. https://doi.org/10.1007/978-0-8176-4573-1