MR2351216 Yoshiyuki KAGEI Yoshiyuki KAGEI Funkcialaj Ekvacioj. Serio Internacia 50 2007 287--337 http://fe.math.kobe-u.ac.jp/FE/FullPapers/50-2/50_287.pdf http://www.ams.org/mathscinet-getitem?mr=MR2351216 The resolvent problem of the linearized compressible Navier-Stokes equation around a given constant state is considered in an infinite layer $\bf{R}^{n-1}\times(0,a)$, $n\geq2$, under the no slip boundary condition for the momentum. It is proved that the linearized operator is sectorial in $W^{1,p}\times L^p$ for $1<p<\infty$. The $L^p$ estimates for the resolvent are established for all $1\leq p\leq \infty$. The estimates for the high frequency part of the resolvent are also derived, which lead to the exponential decay of the corresponding part of the semigroup. Compressible Navier-Stokes equation, Resolvent estimate, Infinite layer. 35Q30, 76N15. 50-287 2007 Yoshiyuki KAGEI Yoshiyuki KAGEI

1 Abe, T.; Shibata, Y. J. Math. Soc. Japan 55 2003 469-497 MR1961297

2 Abe, T.; Shibata, Y. J. Math. Fluid Mech. 5 2003 245-274 MR1994781

3 Abels, H.; Wiegner, M. Differential and Integral Equations 18 2005 1081-1110 MR2162625

4 Beale, J. T.; Nishida, T. Lecture Notes in Num. Appl. Anal. 8 1985 1-14, Recent Topics in Nonlinear PDE II, Sendai, 1985 MR0882925

5 Bergh, J.; Löfström, J. Interpolation Spaces, An Introduction Springer-Verlag, Berlin, Heidelberg, New York 1976 MR0482275 6 Kagei, Y., Asymptotic behavior of the semigroup associated with the linearized compressible Navier-Stokes equation in an infinite layer, to appear in Publ. Res. Inst. Math. Sci. 7 Kagei, Y., Large time behavior of solutions to the compressible Navier-Stokes equation in an infinite layer, preprint, MHF Preprint Series, Kyushu University, MHF 2007-11

8 Kagei, Y.; Kobayashi, T. Arch. Rational Mech. Anal. 165 2002 89-159 MR1946756

9 Kagei, Y.; Kobayashi, T. Arch. Rational Mech. Anal. 177 2005 231-330 MR2188049

10 Kobayashi, T.; Shibata, Y. Pacific J. Math. 207 2002 199-234 MR1974473