Evolution problems in the mechanics of visco-plastic media

We obtain conditions that a Cauchy problem for a differential inclusion arising in the mechanics of visco-plastic media be well posed. We establish the complete continuity of shift operators along the trajectory of the inclusion. We obtain variants of the Kneser–Fukuhara theorem on the structure of an integral vortex and an analogue of the first Bogolyubov theorem on the averaging of differential equations.