On the Existence of Weak Solutions for a Nonlinear System of Atmosphere Dynamics with Humidity and Heat Transfer and the Spectral Properties of the Linear Problem

A nonlinear model of the mathematical fluid dynamics of the Atmosphere is considered. The model is a generalization of the nonlinear Navier-Stokes system with the addition of the equations for changeable density, humidity, moisture content in the clouds and heat transfer. An explicit algorithm for a weak solution is constructed by Galerkin method, the “a priori” estimates for the weak solution are obtained and the proof of the existence of the weak solution is given. We also find a sector of the complex plane to which all the eigenvalues of the spectral linearized problem belong. Keywords - Atmospheric sciences, computational science, Galerkin method, nonlinear partial differential equations.