On Uniqueness For The Three-Dimensional Vlasov-Navier-Stokes System
D Han-Kwan, É Miot, A Moussa, I Moyano
TL;DR
The paper addresses the issue of uniqueness for the 3D Vlasov-Navier-Stokes system by proving weak-strong uniqueness when the fluid velocity lies in the CMP Besov class, surpassing the Osgood uniqueness regime. It develops a stability framework combining decoupled Navier–Stokes and Vlasov estimates, using Lagrangian flows and a $Q$-type trajectory distance to control the coupling, and then extends the analysis to local and global well-posedness in Besov CMP spaces via Chemin’s method and Danchin-type decay. Key contributions include a quantitative stability bound and a Besov-space local well-posedness result that extends to global existence for small data through long-time decay. The results significantly advance the understanding of uniqueness and stability for kinetic-fluid couplings in 3D, providing a robust path to well-posedness in critical spaces and bridging kinetic theory with fluid dynamics.
Abstract
We study the problem of uniqueness of Leray solutions to the three-dimensional Vlasov-Navier-Stokes system. We establish uniqueness whenever the fluid velocity field belongs to the Cannone-Meyer-Planchon class, which allows to go beyond the Osgood uniqueness class. A stability estimate in this setting is also provided.