Global Classical Solution to the Navier-Stokes-Vlasov Equations with Large Initial Data and Reflection Boundary Conditions

The fluid-particle system is studied in this paper. More precisely, we consider the compressible Navier-Stokes equations coupled to the Vlasov equation through the drag force. This model arises from the research of aerosols, sprays or more generically two-phase flows. We work with one-dimensional case of this model, and prove the existence, uniqueness of global classical solution to an initial-boundary value problem with large initial data and reflection boundary conditions. The proof is based on the local existence theorem and the global a priori estimates. More specifically, we show that the density distribution function of particles has compact support, which plays a crucial role in the hardest part of our proof: the estimates of the higher order derivatives of the solution.