Asymptotic Behavior of the Linearized Semigroup at Space-Periodic Stationary Solution of the Compressible Navier–Stokes Equation

The asymptotic behavior of the linearized semigroup at spatially periodic stationary solution of the compressible Navier–Stokes equation in a periodic layer of R n ( n = 2 , 3 ) is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the linearized semigroup is decomposed into two parts; one behaves like a solution of an n - 1 dimensional linear heat equation as time goes to infinity and the other one decays exponentially.