Global well-posedness of large scale moist atmosphere system with only horizontal viscosity in the dynamic equation

TL;DR

This paper establishes the global well-posedness of a large-scale moist atmosphere system with only horizontal viscosity, using advanced mathematical techniques to prove existence and uniqueness of solutions.

Contribution

It introduces a novel approach to prove global existence and uniqueness for a moist atmosphere model lacking vertical viscosity, expanding the mathematical understanding of such systems.

Findings

Proved local existence of solutions in H^1 space.

Established global existence under higher regularity assumptions.

Achieved uniqueness of solutions using monotone operator theory.

Abstract

In order to find a better physical model to describe the large-scale cloud-water transformation and rainfall, we consider a moist atmosphere model consisting of the primitive equations with only horizontal viscosity in the dynamic equation and a set of humidity equations describing water vapor, rain water and cloud condensates. To overcome difficulties caused by the absence of vertical viscosity in the dynamic equation, we get the local existence of $v$ in $H^{1}$ space by combining the viscous elimination method and the $z-$weak solution method and using the generalized Bihari-Lasalle inequality. And then, we get the global existence of $v$ under higher regularity assumption of initial data. In turn, the existence of quasi-strong and strong solutions to the whole system is obtained. By introducing two new unknown quantities appropriately and utilizing the monotone operator theory to overcome difficulties caused by the Heaviside function in the source terms, we get the uniqueness of solutions.