The motion and mass growth of droplets with phase transitions in a homogeneous medium

In this paper, we focus on the motion and mass growth of droplets with phase transitions in a homogeneous medium. We characterize the problem by the unsteady non-isentropic compressible Euler system together with its Radon measure-valued solutions. That is, the gas is described by the regular part of Radon measure, while the droplets are illustrated by the atomic part. The difficulty lies in finding a suitable formulation of the constitutive equation in the sense of measure, such that it is physically meaningful and mathematically reasonable. We overcome it by proposing one, which can express the process of heat release by liquefaction and heat absorption by vaporization. Then we prove the local-in-time and global-in-time existence of a single droplet with different initial data. Also, we analyze the collision of two droplets and deduce the state of the new droplet formed by collisions. This provides a downscaling new approach to investigating the two-phase flows with phase transitions.