Low Mach number limit of strong solutions to the compressible primitive equations with gravity

TL;DR

This paper analyzes the low Mach number limit of strong solutions to the compressible primitive equations with gravity, establishing uniform estimates and rigorously deriving the incompressible limit for general initial data.

Contribution

It constructs uniform estimates for solutions with ill-prepared data and rigorously proves the low Mach number limit to the incompressible primitive equations considering gravity effects.

Findings

Uniform estimates for solutions with ill-prepared initial data.

Rigorous derivation of the incompressible limit system.

Handling of anisotropic effects and gravity in the analysis.

Abstract

In this paper, we explore the low Mach number singular limit of the local-in-time strong solutions to the compressible primitive equations with gravity for general adiabatic coefficient. First we construct the uniform estimate for the solutions to the non-dimensional compressible primitive equations with general ill-prepared initial data. Due to the effects of gravity and the anisotropy of the system, the operator with large coefficient in this model is not explicitly skew-symmetric. Thus, obtaining the uniform estimate requires novel techniques. After that, we investigate rigorously the low Mach number limit of the compressible primitive equations with both well-prepared and ill-prepared initial data. The limiting system is shown to be the incompressible primitive equations with inhomogeneous density that depends on the vertical variable.