Three-scale singular limits with applications to rapidly rotating fluids and the hyperbolization of dispersive systems

We consider singular problems for a general class of quasilinear hyperbolic systems that involve two a priori independent stiff parameters. We argue that such situations may lead to the rapid development of small-amplitude spatial oscillations of small wavelength starting from arbitrarily smooth initial data. Despite this phenomenon we provide sufficient conditions on initial data that secure the uniform control of solutions and show strong convergence in the singular limit of small stiff parameters. We apply our general theory to the rapidly rotating shallow-water system with bottom topography, and to hyperbolic systems stemming from a constraint-relaxation strategy applied to dispersive models for the propagation of water waves, specifically the Benjamin-Bona-Mahony, Boussinesq-Peregrine and Serre-Green-Naghdi equations.