Low-Mach-number Euler equations with solid-wall boundary condition and general initial data

We prove that the divergence-free component of the compressible Euler equations with solid-wall boundary condition converges strongly towards the incompressible Euler equations at the same order as the Mach number. General initial data are considered and are not necessarily close to the divergence-free state. Thus, large amplitude of fast oscillations persist and interact through nonlinear coupling without any dissipative or dispersive mechanism. It is then shown, however, that the contribution from fast oscillations to the slow dynamics through nonlinear coupling is of the same order as the Mach number when averaged in time. The structural condition of a vorticity equation plays a key role in our argument.