Large time asymptotic behavior of the compressible Navier-Stokes   Equations in partial Space-Periodic Domains
  In this paper, we study the large time behavior of the 3-D isentropic compressible Navier-Stokes equation in the partial space-periodic domains, and simultaneously show that the related profile systems can be described by like Navier-Stokes equations with suitable "pressure" functions in lower dimensions. Our proofs are based on the energy methods together with some delicate analysis on the corresponding linearized problems. 
Yin Huicheng, Zhang Lin, Zhu Lu
math.AP