Low stratifucation of the complete Euler system

The present paper takes advantage of the concept of dissipative measure-valued solutions to show the rigorous derivation of the Euler-Boussinesq (EB) system that has been successfully used in various meteorological models. In particular, we show that EB system can be obtained as a singular limit of the complete Euler system. We provide two types of result - firstly, we treat the case of well-prepared initial data for any sufficiently regular bounded domain. Secondly, we use the dispersive estimates for acoustic equation to tackle the case of the illprepared initial data on an unbounded exterior domain.