Abstract:
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.
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