Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system (Q2177510)

Summary: 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 \textit{G. Loeper} [J. Math. Pures Appl. (9) 86, No. 1, 68--79 (2006; Zbl 1111.35045)]) with the use of Hardy's maximal function, in order to obtain some fine Wasserstein-like estimates for the difference of two solutions of the Vlasov equation.