On the existence of stationary solutions to compressible Navier-Stokes equations
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 \textit{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 \textit{H. Beirão da Veiga} [Houston J. Math. (to appear)].
compressible forms of the Navier-Stokes equations
scientific article; zbMATH DE number 4020722