Asymptotic behavior of solutions to the linearized compressible Navier–Stokes equation around a given constant state is considered in an inﬁnite layer ℝ_n_−1 × (0, a), n ≥ 2, under the no slip boundary condition for the momentum. The L__p decay estimates of the associated semigroup are established for all 1 ≤ p ≤ ∞. It is also shown that the time-asymptotic leading part of the semigroup is given by an n − 1 dimensional heat semigroup.
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Yoshiyuki Kagei, Asymptotic Behavior of the Semigroup Associated with the Linearized Compressible Navier–Stokes Equation in an Inﬁnite Layer. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, pp. 763–794