Anelastic Limits for Euler Type Systems

In this paper, we present rigorous derivations of anelastic limits for compressible Euler type systems when the Mach (or Froude) number tends to zero. The first and main part is to prove local existence and uniqueness of strong solution together with uniform estimates on a time interval independent of the small parameter. The key new remark is that the systems under consideration can be written in a form where ideas from \cite{MS} can be adapted. The second part of the analysis is to pass to the limit as the parameter tends to zero. In this context, the main problem is to study the averaged effect of fast acoustic waves on the slow incompressible motion. In some cases, the averaged system is completely decoupled from acoustic waves. The first example studied in this paper enters this category: it is a shallow-water type system with topography and the limiting system is the inviscid lake equation (rigid lid approximation). This is similar to the low Mach limit analysis for prepared data, following the usual terminology, where the acoustic wave disappears in a pure pressure term for the limit equation. The decoupling also occurs in infinite domains where the fast acoustic waves are rapidly dispersed at infinity and therefore have no time to interact with the slow motion (see \cite{Sc,MS, Al}). In other cases, and this should be expected in general for bounded domains or periodic solutions, this phenomenon does not occur and the acoustic waves leave a nontrivial averaged term in the limit fluid equation, which cannot be incorporated in the pressure term. In this case, the limit system involves a fluid equation, coupled to a nontrivial infinite dimensional system of differential equations which models the energy exchange between the fluid and some remanent acoustic energy. This was suspected for the periodic low Mach limit problem for nonisentropic Euler equations in \cite{MeSc} and proved for finite dimensional models. The second example treated in this paper, namely Euler type system with heterogeneous barotropic pressure law, is an example where this scenario is rigorously carried out. To the authors' knowledge, this is the first example in the literature where such a coupling is mathematically justified.