Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 3, pp. 267-299 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the initial boundary value problem for the linked Navier–Stokes/Fokker–Planck/Poisson equations describing the flow of a viscous incompressible fluid with highly dispersed infusion of solid charged particles which are subjected to a random impact from thermal motion of the fluid molecules. We prove the existence of global weak solutions for the problem and study some properties of these solutions.
@article{JMAG_2014_10_3_a0,
     author = {O. Anoshchenko and S. Iegorov and E. Khruslov},
     title = {Global {Weak} {Solutions} of the {Navier{\textendash}Stokes/Fokker{\textendash}Planck/Poisson} {Linked} {Equations}},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {267--299},
     year = {2014},
     volume = {10},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2014_10_3_a0/}
}
TY  - JOUR
AU  - O. Anoshchenko
AU  - S. Iegorov
AU  - E. Khruslov
TI  - Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2014
SP  - 267
EP  - 299
VL  - 10
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/JMAG_2014_10_3_a0/
LA  - en
ID  - JMAG_2014_10_3_a0
ER  - 
%0 Journal Article
%A O. Anoshchenko
%A S. Iegorov
%A E. Khruslov
%T Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2014
%P 267-299
%V 10
%N 3
%U http://geodesic.mathdoc.fr/item/JMAG_2014_10_3_a0/
%G en
%F JMAG_2014_10_3_a0
O. Anoshchenko; S. Iegorov; E. Khruslov. Global Weak Solutions of the Navier–Stokes/Fokker–Planck/Poisson Linked Equations. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2014) no. 3, pp. 267-299. http://geodesic.mathdoc.fr/item/JMAG_2014_10_3_a0/

[1] A. I. Grigor'ev, T. I. Sidorova, “Some Laws Governing the Settling and Accumulation of an Industrial Aerosol Over a Region”, Techn. Phys., 43:3 (1998), 283–287 | DOI

[2] C. W. Gardiner, Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences, Springer-Verlag, Berlin, 1983 | MR | Zbl

[3] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland Publishing Co., Amsterdam–New York, 1981 | MR

[4] K. Hamdache, “Global Existence and Large Time Behaviour of Solutions for the Vlasov–Stokes Equations”, Japan J. Indust. Appl. Math., 15:1 (1998), 51–74 | DOI | MR

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

[6] O. Anoshchenko, E. Khruslov, H. Stephan, “Global Weak Solutions to the Navier–Stokes–Vlasov–Poisson System”, J. Math. Phys., Anal., Geom., 6:2 (2010), 143–182 | MR | Zbl

[7] S. Egorov, E. Ya. Khruslov, “Global Weak Solutions of the Navier–Stokes–Fokker–Planck System”, Ukrainian Math. J., 65:2 (2013), 212–248 | DOI | MR | Zbl

[8] A. A. Arsenev, “Existence in the Large of a Weak Solution of Vlasov's System of Equations”, Zh. Vycisl. Mat. i Mat. Fiz., 15 (1975), 136–147 | MR | Zbl

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

[10] P. Degond, “Global Existence of Smooth Solutions for the Vlasov–Fokker–Planck Equation in 1 and 2 Space Dimensions”, Ann. Sci. École Norm. Sup., 19:4 (1986), 519–542 | MR | Zbl

[11] K. Pfaffelmoser, “Global Classical Solutions of the Vlasov–Poisson System in Three Dimensions for General Initial Data”, J. Differ. Eqs., 95:2 (1992), 281–303 | DOI | MR | Zbl

[12] R. Alexandre, “Weak Solutions of the Vlasov–Poisson Initial-Boundary Value Problem”, Math. Methods Appl. Sci., 16:8 (1993), 587–607 | DOI | MR | Zbl

[13] F. Bouchut, “Existence and Uniqueness of a Global Smooth Solution for the Vlasov–Poisson–Fokker–Planck System in Three Dimensions”, J. Funct. Anal., 111:1 (1993), 239–258 | DOI | MR | Zbl

[14] J. A. Carrillo, J. Soler, “On the Initial Value Problem for the Vlasov–Poisson–Fokker–Planck System with Initial Data in $L^p$ Spaces”, Math. Methods Appl. Sci., 18:10 (1995), 825–839 | DOI | MR | Zbl

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

[16] A. I. Košelev, “A Priori Estimates in $L_p$ and Generalized Solutions of Elliptic Equations and Systems”, Uspehi Mat. Nauk, 13:4 (82) (1958), 29–88 | MR

[17] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris; Gauthier-Villars, 1969 | MR | Zbl

[18] O. Anoshchenko, O. Lysenko, E. Khruslov, “On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems”, J. Math. Phys., Anal., Geom., 5:2 (2009), 115–122 | MR | Zbl

[19] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Nauka, M., 1966 | MR | Zbl