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We study a moisture model for warm clouds that has been used by Klein andMajda as a basis for multiscale asymptotic expansions for deep convectivephenomena. These moisture balance equations correspond to a bulk microphysicsclosure in the spirit of Kessler and of Grabowski and Smolarkiewicz, in whichwater is present in the gaseous state as water vapor and in the liquid phase ascloud water and rain water. It thereby contains closures for the phase changescondensation and evaporation, as well as the processes of autoconversion ofcloud water into rainwater and the collection of cloud water by the fallingrain droplets. Phase changes are associated with enormous amounts of latentheat and therefore provide a strong coupling to the thermodynamic equation.  In this work we assume the velocity field to be given and prove rigorouslythe global existence and uniqueness of uniformly bounded solutions of themoisture model with viscosity, diffusion and heat conduction. To guaranteelocal well-posedness we first need to establish local existence results forlinear parabolic equations, subject to the Robin boundary conditions on thecylindric type of domains under consideration. We then derive a prioriestimates, for proving the maximum principle, using the Stampacchia method, aswell as the iterative method by Alikakos to obtain uniform boundedness. Theevaporation term is of power law type, with an exponent in general less orequal to one and therefore making the proof of uniqueness more challenging.However, these difficulties can be circumvented by introducing new unknowns,which satisfy the required cancellation and monotonicity properties in thesource terms.

Global well-posedness for passively transported nonlinear moisture dynamics with phase changes