Global Weak Solutions to Compressible Navier–Stokes–Vlasov–Boltzmann Systems for Spray Dynamics

Abstract

This work focus on the construction of weak solutions to a kinetic-fluid system of partial differential–integral equations modeling the evolution of particles droplets in a compressible fluid. The system is given by a coupling between the standard isentropic compressible Navier–Stokes equations for the macroscopic description of a gas fluid flow, and a Vlasov–Boltzmann type equation governing the evolution of spray droplets modeled as particles with varying radius. We establish the existence of global weak solutions with finite energy, whose density of gas satisfies the renormalized mass equation. The proof combines techniques inspired by the work of Feireisl et al. (J Math Fluid Mech 3:358–392, 2001) on the weak solutions of the compressible Navier–Stokes equations in a coupled system to the kinetic problem for the spray droplets by extending techniques of Leger and Vasseur (J Hyperbolic Differ Equ 6(1):185–206, 2009) developed for the incompressible fluid-kinetic system.