Formulation of the equations of the humid atmosphere in the context of variational inequalities

The equations of the humid atmosphere with saturation are considered in the case where the saturation concentration qsqs is not assumed to be constant and obeys the laws of thermodynamics [22], [23] and [32]. This article starts from the observation that these equations are not correct in the extreme cases where the atmosphere is totally dry or totally wet, that is when the concentration q of water vapor is equal to 0 or 1. A new formulation of the equations is proposed to remedy this difficulty in the context of variational inequalities. Results of existence, maximum principle and regularity of solutions are derived for these new variational inequalities.