Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system

TL;DR

This paper establishes the uniqueness of weak solutions to the 2D Vlasov-Navier-Stokes system in both the whole space and periodic settings, using optimal transportation and Hardy's maximal function to derive Wasserstein-like estimates.

Contribution

It introduces a novel combination of optimal transportation methods and Hardy's maximal function to prove uniqueness for the 2D Vlasov-Navier-Stokes system.

Findings

Proved uniqueness of weak solutions in 2D Vlasov-Navier-Stokes system

Developed Wasserstein-like estimates for solution differences

Applicable in both whole space and periodic cases

Abstract

We prove a uniqueness result for weak solutions to the Vlasov-Navier-Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy's maximal function, in order to obtain some fine Wassestein-like estimates for the difference of two solutions of the Vlasov equation.