Justification of the averaging method for the convection problem with high-frequency vibrations

The averaging method for the convection problem in the field of high-frequency oscillating forces is justified. The main difficulty in such a problem relates to the presence of a high-frequency factor proportional to the frequency $\omega\gg1$, standing on the right-hand side of the Navier–Stokes system. In the present article, this factor is $\omega P(x,\sin\omega t,\cos\omega t)$, where $P(x,t_1,t_2)$ is an arbitrary polynomial in $t_1$ and $t_1$ whose coefficients depend on the space variable $x$ and whose average is
$$ \overline{P}(x)=(2\pi)^{-1}\int_0^{2\pi}P(x,\sin\tau,\cos\tau)\,d\tau=0. $$