Abstract
With solid-wall boundary condition and ill-prepared initial data, we prove the singular limits and convergence rates of compressible Euler and rotating shallow water equations towards their incompressible counterparts. A major issue is that fast acoustic waves contribute to the slow vortical dynamics at order one and do not damp in any strong sense. Upon averaging in time, however, such a contribution vanishes at the order of the singular parameters (i.e., Mach/Froude/Rossby numbers). In particular, convergence rates of the compressible dynamics, when projected onto the slow manifold, are estimated explicitly in terms of the singular parameters and Sobolev norms of the initial data. The structural condition of a vorticity equation plays a key role in such an estimation as well as in proving singular-parameter-independent life spans of classical solutions.