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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.

MSC:
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
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[1] Acerbi, E. and Fusco, N.:An approximation lemma forW1,pfunctions. InMaterial instabilities in continuum mechanics (Edinburgh, 1985-1986),1-5.Oxford Sci. Publ., Oxford Univ. Press, New York, 1988.
[2] Anoshchenko, O. and Boutet de Monvel-Berthier, A.:The existence of the global generalized solution of the system of equations describing suspension motion. Math. Methods Appl. Sci.20(1997), no. 6, 495-519. · Zbl 0871.35075
[3] Bahouri, H., Chemin, J.-Y. and Danchin, R.:Fourier analysis and nonlinear partial differential equations.Grundlehren der Mathematischen Wissenschaften 343, Springer, Heidelberg, 2011. · Zbl 1227.35004
[4] Benjelloun, S., Desvillettes, L. and Moussa, A.:Existence theory for the kinetic-fluid coupling when small droplets are treated as part of the fluid.J. Hyperbolic Differ. Equ.11(2014), no. 1, 109-133. · Zbl 1293.35020
[5] Boudin, L., Desvillettes, L., Grandmont, C. and Moussa, A.:Global existence of solutions for the coupled Vlasov and Navier-Stokes equations.Differential Integral Equations22(2009), no. 11-12, 1247-1271. · Zbl 1240.35403
[6] Boudin, L., Grandmont, C., Lorz, A. and Moussa, A.:Modelling and numerics for respiratory aerosols.Commun. Comput. Phys.18(2015), no. 3, 723-756. · Zbl 1373.76078
[7] Boudin, L., Grandmont, C. and Moussa, A.:Global existence of solutions to the incompressible Navier-Stokes-Vlasov equations in a time-dependent domain. J. Differential Equations262(2017), no. 3, 1317-1340. · Zbl 1371.35214
[8] Boyer, F. and Fabrie, P.:Mathematical tools for the study of the incompressible Navier-Stokes equations and related models.Applied Mathematical Sciences 183, Springer, New York, 2013. · Zbl 1286.76005
[9] Chemin, J.-Y. and Gallagher, I.:On the global wellposedness of the 3-D Navier- Stokes equations with large initial data.Ann. Sci. ´Ecole Norm. Sup. (4)39(2006), no. 4, 679-698. · Zbl 1124.35052
[10] Chemin, J.-Y. and Lerner, N.:Flot de champs de vecteurs non lipschitziens et ´equations de Navier-Stokes.J. Differential Equations121(1995), no. 2, 314-328. · Zbl 0878.35089
[11] Choi, Y.-P. and Kwon,B.:Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations.Nonlinearity28(2015), no. 9, 3309-3336. · Zbl 1326.35241
[12] Crippa, G. and De Lellis, C.:Estimates and regularity results for the DiPerna- Lions flow.J. Reine Angew. Math.616(2008), 15-46. · Zbl 1160.34004
[13] DiPerna, R. J. and Lions, P.-L.:Ordinary differential equations, transport theory and Sobolev spaces.Invent. Math.98(1989), no. 3, 511-547. · Zbl 0696.34049
[14] Glass, O., Han-Kwan, D. and Moussa, A.:The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of regular equilibria.Arch. Rational. Mech. Anal. 230(2018), no. 2, 593-639. · Zbl 1398.35242
[15] Lemari´e-Rieusset,P. G.:Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.
[16] Lions, P.-L. and Perthame, B.:Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system.Invent. Math.105(1991), no. 2, 415-430. · Zbl 0741.35061
[17] Loeper, G.:Uniqueness of the solution to the Vlasov-Poisson system with bounded density.J. Math. Pures Appl. (9)86(2006), no. 1, 68-79. · Zbl 1111.35045
[18] Moyano,I.:Local null-controllability of the 2-D Vlasov-Navier-Stokes system. Preprint, arXiv:1607.05578, 2016.
[19] Stein, E. M.:Singular integrals and differentiability properties of functions.Princeton Mathematical Series 30, Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501
[20] Wang, D. and Yu, C.:Global weak solution to the inhomogeneous Navier-Stokes- Blasov equations.J. Differential Equations259(2015), no. 8, 3976-4008. · Zbl 1320.35295
[21] Yu, C.:Global weak solutions to the incompressible Navier-Stokes-Vlasov equations.J. Math. Pures Appl. (9)100(2013), no. 2, 275-293. · Zbl 1284.35119
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