Global Weak Solutions of the Navier-Stokes/Fokker-Planck/Poisson Linked Equations
DOI:
https://doi.org/10.15407/mag10.03.267Ключові слова:
уравнение Навье-Стокса, уравнение Фоккера-Планка, уравнение Пуассона, глобальное слабое решение, модифицированный метод Галеркина, теорема Шаудера о неподвижной точке, компактность приближений.Анотація
Рассматривается начально-краевая задача для системы связанных уравнений Навье-Стокса/Фоккера-Планка/Пуассона, описывающей течение вязкой несжимаемой жидкости с высокодисперсной примесью твердых заряженных частиц, подверженных случайным воздействиям, обусловленным тепловым движением молекул жидкости. Доказано существование слабых глобальных решений этой задачи и изучены их свойства.Mathematics Subject Classification: 35A01, 35Q30, 35Q84.
Посилання
A.I. Grigor’ev and T.I. Sidorova, Some Laws Governing the Settling and Accumulation of an Industrial Aerosol Over a Region. — Techn. Phys. 43 (1998), No. 3, 283–287.
C.W. Gardiner, Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences. Springer–Verlag, Berlin, 1983.
N.G. van Kampen, Stochastic Processes in Physics and Chemistry. North-Holland Publishing Co., Amsterdam–New York, 1981.
K. Hamdache, Global Existence and Large Time Behaviour of Solutions for the Vlasov–Stokes Equations. — Japan J. Indust. Appl. Math. 15 (1998), No. 1, 51–74.
A. Mellet and A. Vasseur, Global Weak Solutions for a Vlasov–Fokker– Planck/Navier–Stokes System of Equations. — Math. Models Methods Appl. Sci. 17 (2007), No. 7, 1039–1063.
O. Anoshchenko, E. Khruslov, and H. Stephan, Global Weak Solutions to the Navier–Stokes–Vlasov–Poisson System. — J. Math. Phys., Anal., Geom. 6 (2010), No. 2, 143–182.
S. Egorov and E.Ya. Khruslov, Global Weak Solutions of the Navier–Stokes–Fokker– Planck System. — Ukrainian Math. J. 65 (2013), No. 2, 212–248.
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.
J. Schaeffer, Global Existence of Smooth Solutions to the Vlasov–Poisson System in Three Dimensions. — Comm. Part. Differ. Eqs. 16 (1991), Nos. 8–9, 1313–1335.
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 (1986), No. 4, 519–542.
K. Pfaffelmoser, Global Classical Solutions of the Vlasov–Poisson System in Three Dimensions for General Initial Data. — J. Differ. Eqs. 95 (1992), No. 2, 281–303.
R. Alexandre, Weak Solutions of the Vlasov–Poisson Initial-Boundary Value Problem. — Math. Methods Appl. Sci. 16 (1993), No. 8, 587–607.
F. Bouchut, Existence and Uniqueness of a Global Smooth Solution for the Vlasov– Poisson–Fokker–Planck System in Three Dimensions. — J. Funct. Anal. 111 (1993), No. 1, 239–258.
J.A. Carrillo and J. Soler, On the Initial Value Problem for the Vlasov–Poisson– Fokker–Planck System with Initial Data in Lp Spaces. — Math. Methods Appl. Sci. 18 (1995), No. 10, 825–839.
C. Bardos and 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 (1985), No. 2, 101–118.
A.I. Košelev, A Priori Estimates in Lp and Generalized Solutions of Elliptic Equations and Systems. — Uspehi Mat. Nauk 13 (1958), No. 4 (82), 29–88.
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Gauthier–Villars, Paris, 1969.
O. Anoshchenko, O. Lysenko, and E. Khruslov, On Convergence of Solutions of Singularly Perturbed Boundary-Value Problems. — J. Math. Phys., Anal., Geom. 5 (2009), No. 2, 115–122.
M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, and P.E. Sobolevskii, IntegralOperators in Spaces of Summable Functions. Nauka, Moscow, 1966.