Resolvent Estimates for the Linearized Compressible Navier-Stokes Equation in an Infinite Layer

Abstract

The resolvent problem of 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. It is proved that the linearized operator is sectorial in W^1,p × L^p for 1 < p < ∞. The L^p estimates for the resolvent are established for all 1 ≤ p ≤ ∞. 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.