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

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


76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
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[1] Agmons, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Commun. Pure Appl. Math., Part I:12, 623-727 (1959), Part II:17, 35-92 (1964) · Zbl 0093.10401
[2] Beir?o da Veiga, H.: Stationary motions and the incompressible limit for compressible viscous fluids, MRC Technical Summary Report # 2883, Mathematics Research Center, University of Wisconsin-Madison (1985). To appear in Houston J. Math. · Zbl 0663.76066
[3] Beir?o da Veiga, H.: Existence results in Sobolev spaces for a stationary transport equation (to appear) · Zbl 0691.35087
[4] Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Semin. Mat. Univ. Padova31, 308-340 (1961) · Zbl 0116.18002
[5] Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math.34, 481-524 (1981) · Zbl 0476.76068
[6] Majda, A.: Smooth solutions for the equations of compressible and incompressible fluid flow. In: Fluid dynamics. Beir?o da Veiga, H. (ed.), Lecture Notes in Mathematics, Vol.1047, Berlin, Heidelberg, New York: Springer 1984 · Zbl 0543.76097
[7] Matsumura, A., Nishida, T.: Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys.89, 445-464 (1983) · Zbl 0543.76099
[8] Padula, M.: Existence and uniquenness for viscous steady compressible motions (to appear) · Zbl 0644.76086
[9] Schochet, S.: The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit. Commun. Math. Phys.104, 49-75 (1986) · Zbl 0612.76082
[10] Serrin, J.: Mathematical principles of classical fluid mechanics. Handbuch der Physik, Bd. VIII/1, Berlin, G?ttingen, Heidelberg: Springer 1959 · Zbl 0086.20001
[11] Valli, A.: Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. Ann. Sc. Norm. Super. Pisa 607-647 (1984) · Zbl 0542.35062
[12] Valli, A.: On the existence of stationary solutions to compressible Navier-Stokes equations. Preprint U.T.M. 193, Universit? di Trento (1985). To appear in Annales Institute H. Poincar?, Analyse non lin?aire · Zbl 0633.76067
[13] Valli, A., Zajaczkowski, W. M.: Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case. Commun. Math. Physics103, 259-296 (1986) · Zbl 0611.76082
[14] Defranceschi, A.: On the stationary, compressible and incompressible Navier-Stokes equations. To appear in Ann. Mat. Pura Applicata · Zbl 0663.76067
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