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Asymptotic behavior of the semigroup associated with the linearized compressible Navier-Stokes equation in an infinite layer. (English) Zbl 1181.35172
The author considers the linearisation of the Navier-Stokes equations in a steady state posed in an infinite layer $$\mathbb R^m\times (0,a)$$. Using the author’s previous result that this generates an analytic semigroup [Funkc. Ekvacioj, Ser. Int. 50, No. 2, 287–337 (2007; Zbl 1180.35413)], $$L^p$$-decay properties are established for all $$1\leq p \leq \infty$$. In contrast to the mixed hyperbolic-parabolic behaviour for the problem posed in the whole space [D. Hoff and K. Zumbrun, Z. Angew. Math. Phys. 48, No. 4, 597–614 (1997; Zbl 0882.76074)], it is shown that the leading order part of the semigroup for large times is an $$m$$-dimensional heat semigroup.

##### MSC:
 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 76N15 Gas dynamics (general theory)
##### Keywords:
decay estimates; heat semigroup; Navier-Stokes equations
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##### References:
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