Zero Mach Number Limit of the Compressible Primitive Equations Part I: Well-prepared Initial Data

Abstract: This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order $ \mathcal O(\varepsilon) $, as $ \varepsilon \rightarrow 0^+ $, where $ \varepsilon $ represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.