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 $\mathrm{Re}$ vanish, we derive the {\em weakly compressible Stokes approximation} --- a {\em linear} system. When we consider regimes in which the transport scales vanish while $\mathrm{Re}$ maintains order unity, we derive the {\em weakly compressible Navier-Stokes approximation}---a {\em 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 this is the case in a general setting of hyperbolic-parabolic systems that possess an entropy under a structure condition (which is satisfied by the compressible Navier-Stokes system.) Moreover, we obtain a regularity result for the acoustic modes for the weakly compressible Navier-Stokes system.

In the kinetic setting, our derivations begin with the Boltzmann equation. Our work extends earlier derivations of the incompressible Navier-Stokes system by the inclusion of the acoustic modes. We study the validity of these approximations in the setting of the DiPerna-Lions global solutions. Assuming that DiPerna-Lions solutions satisfy the local conservation law of energy, we use a relative entropy method to justify the weakly compressible Stokes approximation which unifies the Acoustic-Stokes limits result of Golse-Levermore, and to justify the weakly compressible Navier-Stokes approximation modulo further assumptions about passing to the limit in certain relative entropy dissipation terms. This last result extends the result of Golse-Levermore--Saint-Raymond for the incompressible Navier-Stokes system.