The problem investigated in this part can be seen as a particular case of the study of the asymptotic behavior (when ε tends to 0) of solutions of systems of the type where Δε is a non-negative operator of order 2 possibly depending on ε, and A is a skew-symmetric operator. This framework contains of course a lot of problems including hyperbolic cases when Δε = 0. Let us notice that, formally, any element of the weak closure of the family (uε)ε>0 belongs to the kernel of A. We can distinguish from the beginning two types of problems depending on the nature of the initial data. The first case, known as the well-prepared case, is the case when the initial data belong to the kernel of A. The second case, known as the ill-prepared case, is the general case. In the well-prepared case, let us mention the pioneer paper by S. Klainerman and A. Majda about the incompressible limit for inviscid fluids. A lot of work has been done in this case. In the more specific case of rotating fluids, let us mention the work by T. Beale and A. Bourgeois and T. Colin and P. Fabrie. In the case of ill-prepared data, the nature of the domain plays a crucial role. The first result in this case was established in 1994 in the pioneering work by S. Schochet for periodic boundary conditions. In the context of general hyperbolic problems, he introduced the key concept of limiting system (see the definition on page 125). In the more specific case of viscous rotating fluids, E. Grenier proved in 1997 in Theorem 6.3, page 125, of this book. At this point, it is impossible not to mention the role of the inspiration played by the papers by J.-L. Joly, G. Métivier and J. Rauch (see for instance and). In spite of the fact that the corresponding theorems have been proved afterwards, the case of the whole space, the purpose of Chapter 5 of this book, appears to be simpler because of the dispersion phenomena.
Streaming patterns in Faraday waves