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

TLDR: In this article, the authors consider the Navier-Stokes equations coupled to the Vlasov equation through the drag force and prove the existence, uniqueness of global classical solution to an initial-boundary value problem with large initial data and reflection boundary conditions.

Abstract: 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.