## Weakly Compressible Navier-Stokes Approximation of Gas Dynamics

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##### Date

2006-08-07##### Author

Jiang, Ning

##### Advisor

Levermore, Charles David

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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.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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