Theoretical basis of the method of successive approximations for stationary problems of the mechanics of a viscous fluid with free separation boundaries

Journal of Mathematical Sciences - Tập 28 - Trang 777-782 - 1985
V. Ya. Rivkind

Tóm tắt

One considers problems with free separation boundaries for flows of a viscous incompressible fluid, described by a complete system of Navier-Stokes equations. One presents a scheme for the construction of approximate methods, giving the possibility to obtain the foundation of energy-type estimates. These schemes are constructed on the basis of a priori estimates, obtained previously at the proof of existence and uniqueness theorems.

Tài liệu tham khảo

O. A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordan and Breach (1969). V. A. Solonnikov and V. E. Shchadilov, “On a certain boundary-value problem for the stationary system of Navier-Stokes equations,” Tr. Mat. Inst. Akad. Nauk SSSR,125, 196–210 (1973). V. Ya. Rivkind, “The investigation of certain problems of the flow of multilayered viscous incompressible fluids,” in: Proceedings of the Seminar on Partial Differential Equations Dedicated to I. G. Petrovskii on his 75th Birthday [in Russian], Moscow (1979), pp. 423–424. V. V. Pukhnachev, “The plane stationary problem with a free boundary for the Navier-Stokes equations,” Zh. Prikl. Mekh. Tekh. Fiz., No. 3, 91–102 (1972). O. A. Ladyzhenskaya and V. G. Osmolovskii, “On the free surface of a layer of fluid over a solid sphere,” Vestn. Leningr. Univ., No. 13, 25–30 (1976). O. A. Ladyzhenskaya and V. E. Rivkind, “Questions of the theory of difference schemes for Navier-Stokes equations and certain results in their numerical solution,” in: Proc. Fourth All-Union Seminar on Numerical Methods in the Mechanics of a Viscous Fluid [in Russian], Novosibirsk (1977), pp. 1–18. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam (1977). V. P. Il'in, “Certain inequalities in functional spaces and their application to the investigation of the convergence of variational processes,” Tr. Mat. Inst. Akad. Nauk SSSR,53, 64–127 (1959).