Vol. 3, No. 1, 2022

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

Daniel Han-Kwan

Vol. 3 (2022), No. 1, 35–67
Abstract

We study the large-time behavior of small-data solutions to the Vlasov–Navier–Stokes system on 3 × 3. 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 HKMM: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, and (2) a bootstrap argument allows us to obtain such a control. This second 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.

Keywords
fluid-kinetic PDEs, large-time behavior
Mathematical Subject Classification
Primary: 35Q83, 76D05
Milestones
Received: 14 August 2020
Revised: 19 April 2021
Accepted: 19 July 2021
Published: 11 May 2022
Authors
Daniel Han-Kwan
Centre de Mathématiques Laurent Schwartz
CNRS & École Polytechnique
Palaiseau
France