Global Existence and Uniqueness of Strong Solutions for a Phase Transition Model in Atmospheric Dynamics

TL;DR

This paper proves the global existence and uniqueness of strong solutions for a phase transition model in atmospheric dynamics, addressing nonlinear discontinuities related to precipitation formation.

Contribution

It introduces a regularized formulation to handle multivalued nonlinearities and establishes existence and uniqueness of solutions without adding viscosity.

Findings

Existence of strong solutions to the original model.

Uniqueness under a physically meaningful assumption.

Rigorous justification of the tropical climate model.

Abstract

In this work, we study a phase transition model in atmospheric dynamics, inspired by the works [6,14,15], which analyze the primitive equations governing the evolution of velocity, temperature, and specific humidity. The main difficulty arises from the presence of a multivalued discontinuous nonlinear term in the temperature and in the humidity equations, describing the formation of precipitations, which becomes active under supersaturation conditions. To overcome this issue, we introduce a regularized formulation that ensures the existence and uniqueness of approximate solutions. By employing classical compactness arguments, we then establish the existence of a strong solution to the original model. Additionally, we establish uniqueness under a conditional and physically meaningful assumption. This approach allows us to provide a rigorous justification of the tropical climate model on the whole space $\mathbb{R}^2$, while avoiding the introduction of a viscosity term in the humidity equation.