Local well-posedness of strong solutions to the non-isentropic compressible primitive equations with vertical diffusion

TL;DR

This paper proves local well-posedness for strong solutions to the non-isentropic compressible primitive equations with vertical diffusion, addressing mathematical challenges due to the absence of a vertical momentum equation.

Contribution

It establishes local existence, uniqueness, and continuous dependence of strong solutions for these equations, which are more complex than classical Navier-Stokes due to derivative loss and nonlinearity.

Findings

Proved local well-posedness for strong solutions

Established regularity requirements for initial data

Addressed mathematical challenges from derivative loss and nonlinearity

Abstract

Due to the absence of dynamical equation in the vertical momentum component of the primitive equations (PEs) of atmospheric dynamics, the vertical component of the velocity can be recovered only from the information on the other physical quantities, while utilizing the hydrostatic balance. This causes one spatial derivative loss while leads to a stronger nonlinearity, comparing to the classic compressible Navier-Stokes equations. As a result, the mathematical analysis on the compressible primitive equations is mathematically more challenging than that on the compressible Navier-Stokes equations. In this paper, we consider the initial-boundary value problem to the non-isentropic compressible primitive equations with only vertical diffusion for the temperature, but without gravity. Local existence and uniqueness as well as the continuous dependence on the initial data of strong solutions are established for any suitably regular initial data. The initial velocity and pressure are assumed here to be of one order derivative higher regularity than that of the initial density.