Two timescales asymptotes of the weakly compressible Stokes system

We establish a two timescale asymptotics of the weakly compressible Stokes system which has dissipation of order ε>0. For any L2 initial data, over time scale of order 1, the solutions of the weakly compressible Stokes system converge strongly to those of the acoustic system as ε→0. Over time scale of order 1/ε, the limit system is the incompressible Stokes system with the initial data projected on the incompressible mode. For the periodic domain, the convergence is weak due to the fast oscillation generated by acoustic waves. For the Navier-slip boundary condition with the reciprocal of slip length being square of the Knudsen number, the acoustic waves are damped by the viscous boundary layer, and consequently the strong convergence is justified.