# Publications of the Research Institute for Mathematical Sciences

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**Volume 51, Issue 2, 2015, pp. 337–372**

**DOI: 10.4171/PRIMS/158**

Published online: 2015-05-11

Spectral Properties of the Linearized Semigroup of the Compressible Navier–Stokes Equation on a Periodic Layer

Yoshiyuki Kagei^{[1]}and Naoki Makio

^{[2]}(1) Kyushu University, Fukuoka, Japan

(2) Toranomon Hills Mori Tower, Tokyo, Japan

The linearized problem for the compressible Navier-Stokes equation around a given constant state is considered in a periodic layer of $\mathbb{R}^{n}$ with $n\geq2$, and spectral properties of the linearized semigroup is investigated. It is shown that the linearized operator generates a $C_0$-semigroup in $L^2$ over the periodic layer and the time-asymptotic leading part of the semigroup is given by a $C_0$-semigroup generated by an $n-1$ dimensional elliptic operator with constant coefficients that are determined by solutions of a Stokes system over the basic period domain.

*Keywords: *compressible Navier–Stokes equation, Stokes system, Bloch wave decomposition, resolvent estimate

Kagei Yoshiyuki, Makio Naoki: Spectral Properties of the Linearized Semigroup of the Compressible Navier–Stokes Equation on a Periodic Layer. *Publ. Res. Inst. Math. Sci.* 51 (2015), 337-372. doi: 10.4171/PRIMS/158