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 from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in ${\mathbf{R}^3}$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in ${L^\infty(]0,\tau^*; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*; W^{1,\infty}(\Omega^+))^2}$ , where ${\Omega^+ = \Omega \times]0, +\infty,\Omega \subset \mathbf{R}^3}$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to ${\partial\Omega}$ ; they lie also in ${W^{1,q}(]0,\tau^*;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*;L^\infty(\Omega^+))^2}$ , where ${q \in 1, \infty]}$ .