Hydrodynamic Limits of the Boltzmann Equation

Abstract

This dissertation studies two problems that are related to the question of how solutions of the Boltzmann equation behave in various fluid dynamic regimes. The Boltzmann equation models so-called rarefied gases of identical particles, for which all but binary collisions between particles can be neglected. When the mean free path of gas particles is small comparing to the macroscopic length scale, one can derive fluid equations from the Boltzmann equations.

The first problem is to establish the acoustic limit for a family of appropriately scaled DiPerna-Lions solutions with finite zeroth to second moments over $\RD$. Every initial data with finite zeroth to second moments has a unique nonhomogeneous global Maxwellian associated with it by matching values of conserved quantities. The fluid fluctuations converge to a unique limit governed by the solution of an acoustic system with variable coefficients. This differs from the acoustic system with constant coefficient obtained by scaling the Boltzmann equation around a homogeneous Maxwellian (cf. Bardos-Golse-Levermore (2000), Golse-Levermore (2002)). Moreover, unlike the regimes around the homogeneous Maxwellian, there is no higher order Navier-Stokes correction in the regime around the nonhomogeneous Maxwellian.

The second problem is the approximation of solutions to the linearized Boltzmann equation by solutions of the linearized compressible Navier-Stokes system and by solutions of the weakly dissipative linearized compressible Navier-Stokes system over a periodic domain. We show that if the initial data of the linearized Boltzmann equation is smooth enough and lies within the fluid regime, then fluid moments of its solutions are close to the associated linearized compressible Navier-Stokes system in $L^2(\TD)$ uniformly for $t> 0$. We also show that solutions of the weakly dissipative linearized compressible Navier-Stokes systems approximate solutions of the linearized compressible Navier-Stokes system uniformly for $t > 0$ in $L^2(\TD)$. Therefore, we justified weakly dissipative linearized compressible Navier-Stokes approximation to the linearized Boltzmann equation. Our work differs from that of Ellis and Pinsky \cite{ellis1975} in that (1) we consider a periodic domain instead of $\RD$, and (2) the collision kernels we consider include those arising from inverse power potentials, as well as the hard sphere case considered in Ellis and Pinsky (1975).