Abstract

This dissertation addresses mathematical issues regarding weakly compressible approximations of gas dynamics that arise both in fluid dynamical and in kinetic settings. These approximations are derived in regimes in which (1) transport coefficients (viscosity and thermal conductivity) are small and (2) the gas is near an absolute equilibrium — a spatially uniform, stationary state. When we consider regimes in which both the transport scales and Re vanish, we derive the weakly compressible Stokes approximation — a linear system. When we consider regimes in which the transport scales vanish while Re maintains order unity, we derive the weakly compressible Navier-Stokes approximation—a quadratic system. Each of these weakly compressible approximations govern both the acoustic and the incompressible modes of the gas. In the fluid dynamical setting, our derivations begin with the fully compressible Navier-Stokes system. We show that the structure of the weakly compressible Navier-Stokes system ensures that it has global weak solutions, thereby extending the Leray theory for the incompressible Navier-Stokes system. Indeed, we show that