Mathematical Problems Arising When Connecting Kinetic to Fluid Regimes

Abstract

In this dissertation we study two problems that are related to the question of how to obtain appropriate macroscopic descriptions of a gas from its microscopic formulation. Mathematically, fluid equations formulate the macroscopic dynamics of a gas while kinetic equations are used to study the microscopic world. One can derive fluid equations from kinetic equations through formal asymptotic expansions like those of Hilbert or Chapman-Enskog. The first problem we study relates to the justification of the steps in those formal expansions, while the second relates to the well-posedness of a resulting fluid system. The first problem we study is that of establishing a Fredholm alternative for the linearized Boltzmann collision operator. The Fredholm alternative is used in both the formal asymptotic derivations and the rigorous justifications of fluid approximations to the Boltzmann equation. Results of this type have been obtained for collision kernels satisfying the Grad angular cutoff assumption. However, because DiPerna-Lions' renormalized solutions for the Boltzmann equation are established for more general collision kernels, it is interesting to extend the Fredholm property of the linearized Boltzmann operator to these collision kernels. We show that under a weak cutoff assumption, the linearized Boltzamnn operator does satisfy the Fredholm alternative. The second problem we study is the well-posedness of a dispersive fluid system that is formally obtained via an asymptotic expansion of the Boltzmann equation as a first correction to the compressible Navier-Stokes system. This system is degenerate in both dissipation and dispersion. Therefore the theory for strictly dispersive systems does not apply directly. To prove the well-posedness of this degenerate system, we need to study different smoothing effects for different components of the solution. We show that using the regularization effects including dispersion and dissipation, this system has a unique smooth solution for a finite time.