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Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case. (English) Zbl 0611.76082
The authors consider the equations which describe the motion of a viscous compressible and heat-conductive fluid, taking into consideration the case of inflow and/or outflow through the boundary.
First, they prove the existence of a global (in time) unique solution under the usual assumption that the data of the problem are small enough and assuming also that on the boundary there is no inflow. In the presence of inflow, it is shown that a global solution (with bounded and strictly positive density) cannot be found using their method. Second, they prove the asymptotic equivalence of solutions which initially have the same amount of mass, in the case of no outflow. Finally, they prove the existence of periodic and stationary solutions in the case of vanishing normal component of the velocity on the boundary.
The previous stability result is an essential tool for proving the existence of these solutions. In the presence of only inflow or only outflow it is shown that periodic and stationary solutions cannot exist. The case of inflow and outflow at the same time is an open problem. The paper is a generalization of the results proved in Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 607-647 (1983; Zbl 0542.35062) by the first author.
Reviewer: P.Secchi

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