Valli, Alberto Navier-Stokes equations for compressible fluids: global estimates and periodic solutions. (English) Zbl 0601.35094 Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math 45, Pt. 2, 467-476 (1986). [For the entire collection see Zbl 0583.00018.] Some results concerning the flow of a barotropic compressible viscous fluid are presented. In the first part, some previous result of the author are presented. The main result is the existence of the global solutions near equilibrium, in the case of time-dependent external forces, which are not square- integrable in t over \((t_ 0,\infty)\), \(t_ 0\) being the initial time. To obtain the existence for large time, a successive application of local existence theorem is used. The proof of the a priori estimates is related in detail; the crucial point is to have a control on the norm of div \(\vec v\) in the Sobolev space \(W^ 1_ 2.\) In the case of ”small” time-periodic forces, the existence of a time- periodic solution is proved. In the last part, the asymptotic stability and the existence of a stationary solution are studied. Reviewer: G.Pasa Cited in 3 Documents MSC: 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B10 Periodic solutions to PDEs 35B45 A priori estimates in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:Navier-Stokes equations; barotropic compressible viscous fluid; existence; global solutions near equilibrium; a priori estimates; Sobolev space; time-periodic solution; asymptotic stability; stationary solution Citations:Zbl 0583.00018 PDFBibTeX XML