Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions

In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent looking for classical solutions in higher order Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima, Atti dell'Accademia delle Scienze di Torino, 2011] as a model for air motion in $R^3$ including water phase transitions. Unknown functions are: the densities $\rho$ of dry air, $\pi$ of water vapor, $\sigma$ and $\nu$ of water in the liquid and solid state, dependent also on the mass $m$ of the droplets or ice particles. Solutions are Lipschitz continuous with respect to the space and mass variables, and also have some regularity in time; they depend continuously on initial data, temperature and velocities.