  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.
