On the stationary, compressible and incompressible Navier-Stokes equation (Q1115276)

We study the system (*): \[ -\mu \Delta u-\nu \nabla div u+\nabla p(\rho,\zeta)=\rho [f-(u\cdot \nabla)u],\quad div(\rho u)=g, \] \[ -\chi \Delta \zeta +c_ v\rho u\cdot \nabla \zeta +\zeta p'_{\zeta}(\rho,\zeta)div u=\rho h+\psi (u,u),\quad in\quad \Omega, \] \[ u|_{\Gamma}=0,\quad \zeta |_{\Gamma}=\zeta_ 0, \] in a bounded open domain \(\Omega\) in \(R^ n\), for arbitrarily large \(n\geq 2\). It is assumed that \(\Omega\) lies (locally) on one side of its boundary \(\Gamma\), a \(C^{j+3}\) manifold. Here, \(\psi (u,u)=\chi_ 0\sum^{n}_{i,j=1}(\partial u_ i/\partial x_ j)+(\partial u_ j/\partial x_ i)^ 2+\chi_ 1(div u)^ 2\), and \((v\cdot \nabla)u=\sum^{n}_{i=1}v_ i(\partial u/\partial x_ i)\). System (*) describes the stationary motion of a compressible, heat conductive, viscous fluid. We consider the incompressible limit of the solutions of that system of equations (for barotropic flows) as the Mach number becomes small.