Hydrodynamic limit for compressible Navier-Stokes-Vlasov-Poisson equations with local alignment force

TL;DR

This paper proves that solutions to the compressible Navier-Stokes-Vlasov-Poisson equations with local alignment force converge to a two-phase fluid model in the hydrodynamic limit, despite challenges due to lack of dissipation in the particle equation.

Contribution

It establishes the hydrodynamic limit for these equations using the relative entropy method, showing convergence to a two-phase fluid model and the Dirac distribution in velocity.

Findings

Distribution function converges to a Dirac distribution in velocity.

Fluid density and velocity converge to smooth solutions.

Global weak solutions approach the limiting two-phase fluid model.

Abstract

We investigate the hydrodynamic limit of weak solutions to compressible Navier-Stokes-Vlasov-Poisson equations with local alignment force in three-dimensional torus domain. Due to the absence of dissipation terms in particle equation, it is difficult to study this problem. Based on the relative entropy method, it is shown that the global weak solutions of the compressible Navier-Stokes-Vlasov-Poisson equations converge to the smooth solutions of the limiting two-phase fluid model.We obtained that the distribution function $f^{\epsilon}$ converges to a Dirac distribution in velocity, the fluid density $\rho^{\epsilon}$ and velocity $u^{\epsilon}$ converge to $\rho$ and $u$, respectively.