The influence of viscosity on oscillations in some linearized problems of hydrodynamics

We establish results on the asymptotic behaviour of solutions of non-stationary linearized equations of hydrodynamics with a small viscosity coefficient and periodic data oscillating rapidly with respect to the spatial variables. We obtain boundary-layer terms, homogenized (limiting) equations and cell problems (whose solutions determine approximate asymptotics of solutions of the equations under consideration) and obtain estimates for the accuracy of the asymptotics. The form of the asymptotics depends strongly on the mutual asymptotic behaviour of the viscosity coefficient and the periodicity parameter that characterizes rapid oscillations of the data. When the viscosity coefficient is very small, the asymptotics can contain rapidly oscillating terms that increase linearly with respect to the time variable. Similar theorems are proved for non-stationary Stokes equations and partial results are obtained for non-stationary Navier–Stokes equations.