Moisture dynamics with phase changes coupled to heat-conducting, compressible fluids

Abstract

It is shown that a model coupling the heat-conducting compressible Navier-Stokes equations to a micro-physics model of moisture in air is locally strongly well-posed for large data in suitable function spaces and strongly well-posed on $[0,\ddot{I}„]$ for every $\ddot{I}„>0$ for small initial data. This seems to be the first result on $[0,\ddot{I}„]$ for arbitrary $\ddot{I}„>0$ for a model coupling moisture dynamics to heat-conducting, compressible Navier-Stokes equations. A key feature of the micro-physics model is that it also includes phase changes of water in moist air. These phase changes are associated with large amounts of latent heat and thus result in a strong coupling to the thermodynamic equation. The well-posedness results are obtained by means of a Lagrangian approach, which allows to treat the hyperbolicity in the continuity equation. More precisely, optimal ${\mathrm{L}}^p$-${\mathrm{L}}^q$ estimates are shown for the linearized system, leading to the local well-posedness result by a fixed point argument and suitable nonlinear estimates. For the well-posedness result on $[0,\ddot{I}„]$ for arbitrary $\ddot{I}„>0$, a refined analysis of the linearized problem close to equilibria is carried out, and the roughness of the source term, induced by the phase changes, requires to establish delicate a priori bounds.

Accepted for publication in Mathematical Models and Methods in Applied Sciences