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\(L^ 2\) decay for the compressible Navier-Stokes equations in unbounded domains. (English) Zbl 0798.35124
The author considers the equations of motion for a viscous, compressible, heat-conducting fluid that occupies the complement of a bounded domain in \(\mathbb{R}^ 3\) or the half-space. If the data are sufficiently small, global existence in time has been proved by A. Matsumura and T. Nishida [Commun. Math. Phys. 89, 445-464 (1983; Zbl 0543.76099)]. Such solutions are shown to tend to a stationary state for \(t\to\infty\) in the \(L_ 2\)-norm. If the initial data are chosen in such a way that the solution to the homogeneous linear problem decays like \(t^{- \alpha}\) for some \(\alpha>0\) then the decay of the solution to the nonlinear problem can be estimated by \(t^{-\beta}\), \(\beta= \min(\alpha,1/8)\).

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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