Relative Entropy and Slightly Compressible Navier-Stokes Dynamics of the Boltzmann Equation

Abstract

This paper shows that, in the formal level, the convergence of solutions of Boltzmann equation to solutions of the compressible Navier-Stokes system with small Mach number over the three-dimensional periodic domain \(\mathbb {T}^3\), using the relative entropy method originated from Bardos, Golse, Levermore [Comm. Pure Appl. Math. 46 (1993) 667–753] and Yau [Lett. Math. Phys. 22 (1991) 63–80]. We discuss the evolution of the entropy which is relative to the local Maxwellian governed by the solution of slightly compressible Navier-Stokes system. This characterizes the convergence rate from Boltzmann equation to the incompressible Navier-Stokes system.