1. Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains
- Author
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Lucas Ertzbischoff, Ayman Moussa, Daniel Han-Kwan, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,01 natural sciences ,Domain (mathematical analysis) ,Exponential function ,010101 applied mathematics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Distribution function ,Flow velocity ,Bounded function ,Dirichlet boundary condition ,FOS: Mathematics ,symbols ,Boundary value problem ,0101 mathematics ,[MATH]Mathematics [math] ,Absorption (electromagnetic radiation) ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We study the large time behaviour of small data solutions to the Vlasov–Navier–Stokes system set on Ω × R 3 , for a smooth bounded domain Ω of R 3 , with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to 0 while the distribution function concentrates towards a Dirac mass in velocity centred at 0, with an exponential rate. The proof, which follows the methods introduced in Han-Kwan et al (2020 Arch. Ration. Mech. Anal. 236 1273–323), requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviours for the kinetic density, from total absorption to no absorption at all.
- Published
- 2021
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