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Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space. (English) Zbl 1038.35057
Summary: We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier-Stokes equation in the half space. The main concern is to analyze the phenomena that happen when the fluid blows out through the boundary. Thus, it is natural to consider the problem in Eulerian coordinates.
We present a necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if the initial perturbation is small in a suitable Sobolev space. The second result is proved by using an \(L^2\)-energy method with the aid of the Poincaré type inequality.

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