An \(L^ p\)-theory for the \(n\)-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions. (English) Zbl 0621.76074

Beirão da Veiga, Hugo

Commun. Math. Phys. 109, 229-248 (1987).

The author studies the system \[ -\mu \Delta u-\nu \nabla div u+\nabla p(\rho,\xi)=\rho [f-(u\cdot \nabla)u],\quad div(\rho u)=g, \]
\[ -\chi \Delta \xi +c_{\nu}\rho u\nabla \xi +\xi p'_{\xi}(\rho,\xi)div u=\rho h+\psi (u,u)\quad in\quad \Omega,\quad u|_{\Gamma}=0,\quad \xi |_{\Gamma}=\xi_ 0 \] describing the stationary motion of a given amount of a compressible heat conducting viscous fluid in a bounded domain \(\Omega\) of \({\mathbb{R}}^ n\), \(n\geq 2\). The existence and uniqueness of a solution (u,\(\rho\),\(\xi)\) for small data (f,g,h) is proved. Moreover the stationary solution of the Navier-Stokes equations is the incompressible limit of stationary solutions of the compressible Navier- Stokes equations as the Mach number becomes small.

Reviewer: I.Bock

MSC:

76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30	Navier-Stokes equations

Keywords:

compressible heat conducting viscous fluid; bounded domain; existence; uniqueness; stationary solution; Navier-Stokes equations; incompressible limit; compressible Navier-Stokes equations

References:

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