Valli, Alberto On the existence of stationary solutions to compressible Navier-Stokes equations. (English) Zbl 0627.76080 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 99-113 (1987). The author considers compressible forms of the Navier-Stokes equations (not necessarily barotropic) in a bounded domain \(\Omega \in {\mathbb{R}}^ 3\) under homogeneous boundary conditions in a stationary (sufficiently small) external force field. In an earlier work [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 607-647 (1983; Zbl 0542.35062)] he proved the existence of stationary solutions by a limiting procedure, starting from the time dependent case. The present paper offers a new proof, proceeding through a linearization and using the Schauder fixed point theorem. To that extent the proof is analogous to the original one given by J. Leray [J. Math. Pures Appl., IX. Seŕ. 12, 1-82 (1933; Zbl 0006.16702)] in the incompressible case. The reviewer remarks that since that case required no smallness restriction it seems reasonable to ask whether that restriction can ultimately be dispensed with here also. The author points out that still another proof was found concomitantly and independently by H. Beirão da Veiga [Houston J. Math. (to appear)]. Reviewer: R.Finn Cited in 42 Documents MSC: 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q30 Navier-Stokes equations Keywords:compressible forms of the Navier-Stokes equations; bounded domain; homogeneous boundary conditions; linearization; Schauder fixed point theorem Citations:Zbl 0542.35062; Zbl 0006.16702 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [2] Friedrichs, K. O., Symmetric positive linear differential equations, Comm. Pure Appl. Math., t. 11, 333-418 (1958) · Zbl 0083.31802 [3] Lax, P. D.; Phillips, R. S., Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., t. 13, 427-455 (1960) · Zbl 0094.07502 [4] Matsumura, A.; Nishida, T., Initial boundary value problems for the equations of motion of general fluids, (Glowinski, R.; Lions, J. L., Computing methods in applied sciences and engineering, V (1982), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam, New York, Oxford) · Zbl 0505.76083 [6] Serrin, J., Mathematical principles of classicalfluid mechanics, Handbuch der Physik, Bd. VIII/I (1959), Springer Verlag: Springer Verlag Berlin, Göttingen, Heidelberg [7] Spivak, M., A comprehensive introduction to differential geometry, t. 4 (1975), Publish or Perish, Inc.: Publish or Perish, Inc. Boston · Zbl 0306.53001 [8] Temam, R., Navier-Stokes equations. Theory and numerical analysis (1977), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam, New York, Oxford · Zbl 0383.35057 [9] Valli, A., Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Scuola Norm. Sup. Pisa, 4, t. 10, 607-647 (1983) · Zbl 0542.35062 [10] Valli, A.; Zajaczkowski, W. M., Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., t. 103, 259-296 (1986) · Zbl 0611.76082 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.