Large time behavior of solutions to the compressible Navier–Stokes equations around periodic steady states

Abstract This paper shows that the strong solution to the compressible Navier–Stokes equation around spatially periodic stationary solution in a periodic layer of R n ( n = 2 , 3 ) exists globally in time if Reynolds and Mach numbers are sufficiently small. It is proved that the asymptotic leading part of the perturbation is given by a solution to the one-dimensional viscous Burgers equation multiplied by a spatially periodic function when n = 2 , and by a solution to the two-dimensional heat equation multiplied by a spatially periodic function when n = 3 .