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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 $$t^{-\beta}$$, $$\beta= \min(\alpha,1/8)$$.

##### MSC:
 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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