Geodesic
Global weak solutions to the Navier–Stokes–Vlasov–Poisson system
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 6 (2010) no. 2, pp. 143-182 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Navier–Stokes–Vlasov–Poisson system describing the flow of a viscous incompressible fluid containing small solid charged particles. The existence result for weak global solutions of the corresponding boundary value problem is obtained. 
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O. Anoshchenko; E. Khruslov; H. Stephan. Global weak solutions to the Navier–Stokes–Vlasov–Poisson system. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 6 (2010) no. 2, pp. 143-182. http://geodesic.mathdoc.fr/item/JMAG_2010_6_2_a0/

[1] S. L. Soo, Multiphase Fluid Dynamics, Hemisphere Publish. Corp., New York, 1989 | MR | Zbl

[2] A. Fortier, Mechanique of Suspensions, Mir, Moscow, 1971 | Zbl

[3] V. A. L'vov, “Convergence of Solutions of an Initial-Boundary Value Problem for a System of Navier–Stokes Equations in Domains with a Moving Fine-Grained Boundary”, Dokl. Akad. Nauk UkrSSR, Ser. A, 85:7 (1987), 21–24 (Russian) | MR

[4] V. A. L'vov, “An Initial-Boundary Value Problem for the Navier–Stokes Equations in Domains with a Random Boundary”, Dokl. Akad. Nauk UkrSSR, Ser. A, 85:3 (1988), 20–23 | MR

[5] O. A. Anoshchenko, Global Existence of a Generalized Solution of a System of Equations of Motion of a Suspension. Dynamical Systems and Complex Analysis, Naukova Dumka, Kyiv, 1992 (Russian) | MR | Zbl

[6] O. A. Anoshchenko, “Existence and Uniqueness of the Solution of a System of Equations of Motion of a Suspension in Holder Classes”, Mat. Fiz., Anal., Geom., 1 (1994), 31–40 (Russian) | MR | Zbl

[7] O. A. Anoshchenko, A. Boutet de Monvel-Berthier, “The Existence of a Global Generalized Solution of the System of Equations Describing Suspension Motion”, Math. Meth., Appl. Sci., 20 (1997), 495–519 | 3.0.CO;2-O class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[8] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, 2, Gordon and Breach Science Publishers, New York, 1969, 224 pp. | MR

[9] S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Boundary Value Problems in the Mechanics of Inhomogeneous Fluids, Nauka (Sibirsk. Otd.), Novosibirsk, 1983 (Russian) | MR | Zbl

[10] A. A. Arsen'ev, “Existence in the Large of a Weak Solution of Vlasov's System of Equations”, Z. Vychisl. Mat. i Mat. Fiz., 15 (1975), 136–147, 276 (Russian) | MR | Zbl

[11] C. Bardos, P. Degond, “Global Existence for the Vlasov-Poisson Equation in 3 Space Variables with Small Initial Data”, Ann. Inst. H. Poincare Anal. Non Lineaire, 2:2 (1985), 101–118 | MR | Zbl

[12] J. Schaeffer, “Global Existence of Smooth Solutions to the Vlasov–Poisson System in Three Dimensions”, Comm. Part. DiR. Eq., 16:8–9 (1991), 1313–1335 | DOI | MR | Zbl

[13] K. Pfaffelmoser, “Global classical Solutions to the Vlasov–Poisson System in Three Dimensions for General Initial Data”, J. Diff. Eq., 95:2 (1992), 281–303 | DOI | MR | Zbl

[14] J. Batt, G. Rein, “Global Classical Solutions of the Periodic Vlasov–Poisson System in Three Dimensions”, C. R. Acad. Sci. Paris, 313 (1991), 411–416 | MR | Zbl

[15] R. Alexandre, “Weak Solutions of the Vlasov–Poisson Initial Boundary Value Problem”, Math. Meth. Appl. Sci., 16 (1993), 587–607 | DOI | MR | Zbl

[16] K. Hamdache, “Global Existence and Large Time Behaviour of Solutions for the Vlasov–Stokes Equations”, Jap. J. Industr. Appl. Math., 15 (1998), 51–74 | DOI | MR

[17] L. Boudin, L. Desvillettes, C. Grandmont, A. Moussa, “Global Existence of Solutions for the Coupled Vlasov and Navier–Stokes Equations”, Diff. Integr. Eqs., 2009 (to appear) | MR

[18] A. Mellet, A. Vasseur, “Global Weak Solutions for a Vlasov–Fokker–Planck/Navier–Stokes Equations”, Math. Mod. Meth. Appl. Sci., 17:7 (2007), 1039–1063 | DOI | MR | Zbl

[19] A. I. Koshelev, “A Priori Estimates in $L_p$”, Usp. Mat. Nauk, 13:4 (1958), 29–88 | MR

[20] S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Translations of Mathematical Monographs, 7, American Mathematical Society, Providence, R.I., 1963, 239 pp. | MR

[21] J-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Paris, 1969 | MR | Zbl

[22] O. Anoshchenko, E. Khruslov, O. Lysenko, “On Convergence of Solutions of Singularily Perturbated Boundary-value Problems”, J. Math. Phys., Anal., Geom., 5 (2009), 115–122 | MR | Zbl

[23] M. Krasnoselskii, P. Zabreiko, E. Pustilnik, P. Sobolevskii, The Integral Operators in the Spaces of Summarized Functions, Nauka, Moscow, 1966 | Zbl