# Homogenization of Navier-Stokes equations with oscillating data.

We consider initial boundary value problems for unsteady Navier- Stokes equations with periodic data rapidly oscillating with respect to the spatial variables, when the oscillations are zero in mean. The problems are stated in bounded domains that are three-dimensional, for example. The period of data oscillations is specified by a positive small parameter *?*and a viscosity coefficient *?*in equations of the problems can be also considered as a positive parameter. We present estimates of solutions of the problems, which are dependent on relations of certain powers of the parameters *?*and *?*. In general case, the presented estimates for velocity fields are actual whenever the viscosity coefficient *?*is not too small in comparison with *?*2. If the condition is fulfilled, then the relevant solutions are small asymptotically in an energy norm and it characterizes a ”smoothing” property for these solutions. In the case, when the viscosity coefficient has order *?*2, the suitable estimates are derived under assumption that a nonlinearity in equations of the problems is ”small” sufficiently. If the condition is fulfilled, then an asymptotics for velocity fields can contain rapidly oscillating terms.

Taras Shevchenko National University of Kyiv

Gennadiy Sandrakov

E-mail: sandrako@mail.ru