In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in \cite{KM} and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler \cite{Ke} and Grabowski and Smolarkiewicz \cite{GS}. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In \cite{HKLT} we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi \cite{CT}, who succeeded in proving the global solvability of the primitive equations.

中文翻译：

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与具有相变的非线性水分动力学耦合的原始方程的全局适定性

在这项工作中，我们研究了大气原始方程的全局可解性，该方程与温云相变的水分动力学耦合，其中水以水蒸气的形式存在，并以云水和雨水的液态形式存在。该水分模型包含相变冷凝和蒸发的闭包，以及云水自动转化为雨水的过程以及落下的雨滴收集云水的过程。它已被 Klein 和 Majda 在 \cite{KM} 中使用，并且对应于 Kessler \cite{Ke} 和 Grabowski 和 Smolarkiewicz \cite{GS} 精神中的体微物理闭包的基本形式。水分平衡通过与相变相关的潜热与热力学方程密切相关。在\cite{HKLT}中，我们假设速度场是给定的，并严格证明了与热力学方程耦合的水分平衡的均匀有界解的全局存在性和唯一性。在本文中，我们提出了一个完整的潮湿大气流动模型的可解性，其中水分模型与控制速度场的大气动力学的原始方程耦合。为了推导速度场的先验估计，我们使用了 Cao 和 Titi \cite{CT} 的思想，他们成功地证明了原始方程的全局可解性。其中水分模型与控制速度场的大气动力学原始方程耦合。为了推导速度场的先验估计，我们使用了 Cao 和 Titi \cite{CT} 的思想，他们成功地证明了原始方程的全局可解性。其中水分模型与控制速度场的大气动力学原始方程耦合。为了推导速度场的先验估计，我们使用了 Cao 和 Titi \cite{CT} 的思想，他们成功地证明了原始方程的全局可解性。