The averaging method in the convection problem with high-frequency oblique vibrations
In former papers the averaging method was used for studying the problem of the onset of a convection flow in a fluid subject to high-frequency vertical vibrations, oblique vibrations, or vibrations in weightlessness. Mathematical justification of applicability of the method to such problems was carried out for vertical oscillations of a particular form and later for a broad class of vertical oscillations and oscillations in weightlessness. In the present article, we extend the justification to a broad class of vibrations acting in arbitrary directions. The results are exposed for some system of evolution differential equations which involves the above-indicated convection problems (as particular cases). We especially point out that here we essentially improve the earlier results in the framework of a more general problem. For instance, we considerably widen the scale that characterizes smoothness of the initial data of the problem, find out conditions under which (weak) solutions are classical, and study instability of solutions in more detail.
scientific article; zbMATH DE number 1023910