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

TL;DR

This paper rigorously justifies the zero Mach number limit of the compressible primitive equations with well-prepared initial data, showing convergence to incompressible flows at a rate of order epsilon and constructing global solutions close to incompressible flows.

Contribution

It provides a rigorous proof of the zero Mach number limit for compressible primitive equations with well-prepared data and constructs global solutions near incompressible flows.

Findings

Convergence rate of order epsilon as Mach number approaches zero.

Identification of incompressible primitive equations as the limit.

Construction of global solutions close to incompressible flows.

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.