ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATION AROUND A TIME-PERIODIC PARALLEL FLOW

The global in time existence of strong solutions to the compressible Navier-Stokes equation around time-periodic parallel flows in R-n, n >= 2, is established under smallness conditions on Reynolds number, Mach number, and initial perturbations. Furthermore, it is proved for n = 2 that the asymptotic leading part of solutions is given by a solution of the one-dimensional viscous Burgers equation multiplied by the time-periodic function. In the case n >= 3 the asymptotic leading part of solutions is given by a solution of the n - 1-dimensional heat equation with the convective term multiplied by the time-periodic function.