Large time behavior of small data solutions to the Vlasov-Navier-Stokes system on the whole space

2020/06/17 by Daniel Han-Kwan, Han-Kwan, Daniel · 2 citations
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Lattice Boltzmann Simulation Studies #Reservoir Engineering and Simulation Methods

paper · doi:10.48550/arxiv.2006.09848

openalex publication_date 2020/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system on \R³ × \R³. We prove that the kinetic distribution function concentrates in velocity to a Dirac mass supported at 0, while the fluid velocity homogenizes to 0, both at a polynomial rate. The proof is based on two steps, following the general strategy laid out in \citeHKMM: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, (2) a bootstrap argument allows to obtain such a control. This last step requires a fine understanding of the structure of the so-called Brinkman force, which follows from a family of new identities for the dissipation (and higher versions of it) associated to the Vlasov-Navier-Stokes system.

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• The incompressible inhomogeneous Navier-Stokes-Vlasov-Fokker-Planck equations: global well-posedness and inviscid limit
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