Mathematical models are often expressed in the form of hyperbolic or hyperbolic-parabolic systems of PDE describing the propagation of non-linear waves. When some parameters governing the behaviour of the equations becomes small, It may happen that the propagation speed of some families of waves becomes much larger than the celerity of the other waves.
In these cases, a formal asymptotic analysis usually predicts that the solutions of the original PDE are close to the solution of a reduced system. For instance, in fluid dynamics we expect that for low Mach number, the solutions are close to the solutions of the incompressible Euler or Navier-Stokes equations.
In this talk, I will review these problems and give some rigorous results for the analysis of reduced MHD models showing that the solutions of the compressible MHD system can indeed be splitted into a high frequency component and a low frequency one described by an incompressible model.