# On weak-strong uniqueness and singular limit for the compressible Primitive Equations

## TL;DR

This paper proves that weak solutions to the compressible Primitive Equations are unique when strong solutions exist and demonstrates the convergence of compressible to incompressible inviscid Primitive Equations in low Mach and high Reynolds number regimes.

## Contribution

It establishes the weak-strong uniqueness property and the singular limit from compressible to incompressible Primitive Equations, bridging a gap in the theoretical understanding.

## Key findings

- Weak-strong uniqueness of solutions established.
- Convergence from compressible to incompressible Primitive Equations proved.
- First demonstration of the link between compressible and incompressible inviscid PE.

## Abstract

This paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.