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Stationary compressible fluids

In J. Serrin Mathematical principles of classical fluid mechanics, In Handbuch der Physik, Bd. VIII/I, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1950. we find the following formulation of a stationary viscous compressible barotropic fluid for the motion in $\R^3$ or $\R^2$. : $$\begin{eqnarray*}&&\rho[(v\cdot \nabla)v-f]-\mu\Delta -(\zeta+\mu/3)\nabla{\rm ......x{in}\quad \R^n\\&&{\rm div}[\rho v]=0\quad\mbox{in} \quad\R^n\end{eqnarray*}$$ The first of these equations corresponds to the conservation of momentum and the second to the conservation of mass. Here $\rho$ is the density of the fluid and $v$ its velocity. $p$ is the pressure. Mostly it is assumed to be a increasing function of $\rho$. With $f$ we assign a external force field. The constants $\mu$ and $\zeta$ are viscosity coefficients. Solutions could be found in papers by Valli in the case of a bounded domain, small Mach numbers and space periodic external force. A corresponding linearized problem is given by $$\begin{eqnarray*}&&-\mu \Delta w - (\zeta+\mu/3)\nabla{\rm div}\; w+(\rho_0v_0\......F\\&&\rho_0{\rm div}\; w+(v_0\cdot \nabla)+{\rm div}(\eta v)=G\end{eqnarray*}$$ Here $\rho_0$ is a positive constant, $p_1>0$ a constant, $v_0=(\omega,0,0)$ and $\omega$ is also a positive constant. There are existence results also in case of high Mach numbers.
 

A late reference: J. L. Boldrini (1990) Stationary Spatially Periodic Compressible Flows at High Mach Number, Rend. Sem. Mat. Univ. Padova, Vol 84.