The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

Abstract

In this article we propose a new formulation of the equations of the humid atmosphere with a multi-phase saturation generalizing thus the model studied in Temam and Wu (2015 J. Funct. Anal. 269 2187–221) and Temam and Wang 2015 System Modeling and Optimization (New York: Springer). More precisely, we consider the more realistic situation where the humid atmosphere comprises three components instead of only one (besides the air), namely water vapor, liquid water and cloud condensates. Furthermore the saturation concentration is not anymore assumed to be constant as it was in Temam and Wu (2015 J. Funct. Anal. 269 2187–221), Temam and Wang 2015 System Modeling and Optimization (New York: Springer). With the additional constraint that the vapor mass ratio is less than the saturation concentration , which depends itself on the state, we are led, from the mathematical point of view, to introduce and handle a system of equations and inequations involving some quasi-variational inequalities for which we prove the existence of solutions.