  We study a moisture model for warm clouds that has been used by Klein and
Majda as a basis for multiscale asymptotic expansions for deep convective
phenomena. These moisture balance equations correspond to a bulk microphysics
closure in the spirit of Kessler and of Grabowski and Smolarkiewicz, in which
water is present in the gaseous state as water vapor and in the liquid phase as
cloud water and rain water. It thereby 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. Phase changes are associated with enormous amounts of latent
heat and therefore provide a strong coupling to the thermodynamic equation.
  In this work we assume the velocity field to be given and prove rigorously
the global existence and uniqueness of uniformly bounded solutions of the
moisture model with viscosity, diffusion and heat conduction. To guarantee
local well-posedness we first need to establish local existence results for
linear parabolic equations, subject to the Robin boundary conditions on the
cylindric type of domains under consideration. We then derive a priori
estimates, for proving the maximum principle, using the Stampacchia method, as
well as the iterative method by Alikakos to obtain uniform boundedness. The
evaporation term is of power law type, with an exponent in general less or
equal 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 the
source terms.
