Global derivation of a Boussinesq-Navier-Stokes type system from fluid-kinetic equations

TL;DR

This paper derives a Boussinesq-Navier-Stokes type system from fluid-kinetic equations over large times using boundary absorption effects, advancing understanding of hydrodynamic limits in half-space geometries.

Contribution

It provides a rigorous derivation of a Boussinesq-Navier-Stokes system from the Vlasov-Navier-Stokes equations for large times in a half-space setting, addressing a previously open question.

Findings

Established decay estimates via boundary absorption effects.

Derived the Boussinesq-Navier-Stokes system from fluid-kinetic equations.

Extended the hydrodynamic limit results to arbitrarily large times.

Abstract

We study a hydrodynamic limit of the Vlasov-Navier-Stokes system with external gravity force, following the framework introduced by Han-Kwan and Michel in [Han-Kwan - Michel, arXiv:2103.06668]. We answer a question raised in the latter concerning the derivation of a Boussinesq-Navier-Stokes type system for arbitrarily large times, starting from the previous fluid-kinetic system. To do so, we consider a particular spatial geometric setting corresponding to the half-space case. Our proof is based on an absorption effect at the boundary which leads to crucial decay in time estimates.