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Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system. (English) Zbl 1437.35536
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 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.

35Q30 Navier-Stokes equations
35Q83 Vlasov equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76N06 Compressible Navier-Stokes equations
76T99 Multiphase and multicomponent flows
35D30 Weak solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
Full Text: DOI
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