Asymptotic behavior of solutions to fluid dynamical equations

This thesis deals with the problem of the asymptotic behavior of solutions to several nonlinear equations from fluid dynamics on both mesoscopic and macroscopic levels, including Boltzmann equation, compressible Navier-Stokes equations and the system of viscous conservation laws with positive definite viscosity matrix. The main purpose is to study the asymptotic behavior of solutions to those equations towards linear and nonlinear waves, such as shock waves, rarefaction waves and contact discontinuities as either the times goes to infinity, or the viscosity and heat conductivity go to zero for the macroscopic equations or the mean free path goes to zero for the mesoscopic equations. Those limit processes are singular. For the system of viscous conservation laws, we show the large time asymptotic nonlinear stability of a superposition of viscous shock waves and viscous contact waves for the system of viscous conservation laws with small initial perturbations, provided that the strengths of these viscous waves are small and of the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a new estimate on the heat equation. For the Boltzmann equation, the main purpose is to study the asymptotic equivalence for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered-rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t = 0) as the mean free path epsilon &rarr 0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. As a by-product, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat-conductivity (depending on the viscosity) is also obtained.