Abstract
We study a class of stationary variational inequalities for Navier--Stokes type operators that can be used to represent problems with nonlinear boundary conditions for equations of motion of viscous fluids. The main result (the solvability theorem) is used for studying one-sided boundary-value problems for equations of heat convection of viscous fluids.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
J.-L. Lions, Quelques méthodes de résolution des problémes auxl imites nonlinéaires, Dunod, Paris, 1969.
A. V. Kazhikhov, “Solvability of several one-sided boundary-value problems for Navier-Stokes equations,” in: Continuum Dynamics [in Russian], no. 16, Novosibirsk, 1974, pp. 5–34.
S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary-Value Problems in Mechanics of Inhomogeneous Fluids [in Russian], Nauka, Novosibirsk, 1983.
A. Yu. Chebotarev, “One-sided and extremum problems related to the Stokes equations,” in: Continuum Dynamics [in Russian], no. 102 Novosibirsk, 1991, pp. 133–147.
A. Yu. Chebotarev, “Subdifferential boundary-value problems for stationary Navier-Stokes equations,” Differentsial′nye Uravneniya [Differential Equations], 28 (1992), no. 8, 1443–1450.
H. Brezis, “Inéquations variationnelles relatives à l'opérateur de Navier-Stokes,” J. Math. Anal. Appl., 39 (1972), no. 1, 159–165.
M. Müller and J. Naumann, “On evolution inequalities of a modified Navier-Stokes type. I, II, III,” Aplikace Matematiky, 24 (1978), no. 2, 174–184, no. 6, 397-407; 24 (1979), no. 2, 81-91.
A. Yu. Chebotarev, “Solvability of the stationary one-sided flow problem for ideal fluids,” in: Continuum Dynamics [in Russian], Novosibirsk, 1987, pp. 129–135.
A. Yu. Chebotarev, “Stationary variational inequalities in the model of inhomogeneous incompressible fluids,” Sibirsk. Mat. Zh. [Siberian Math. J.], 38 (1997), no. 5, 1184–1193.
M. F. Ukhovskii and V. I. Yudovich, “Equations of stationary convection,” Prikl. Mat. Mekh. [J. Appl. Math. Mekh.], 27 (1963), no. 2, 295–300.
V. I. Yudovich, “Origin of convection,” Prikl. Mat. Mekh. [J. Appl. Math. Mekh.], 30 (1966), no. 6, 1000–1005.
V. I. Yudovich, “Free convection and branching,” Prikl. Mat. Mekh. [J. Appl. Math. Mekh.], 31 (1967), no. 1, 101–111.
A. G. Zarubin, “Problem of stationary heat convection,” Zh. Vychisl. Mat. i Mat. Fiz. [U.S.S.R. Comput. Math. and Math. Phys.], 8 (1968), no. 6, 1378–1383.
A. G. Z arubin and M. F. Tiunchik, “Several problems in mechanics with discontinuous boundary conditions and nonsmooth boundary,” Differentsial′nye Uravneniya [Differential Equations], 14 (1978), no. 9, 1632–1637.
G. V. Alekseev, Theoretical Analysis of Stationary Problems of Boundary Control for Equations of Heat Convection [in Russian], Preprint of Institute for Applied Mathematics, Far East Division of the Russian Academy of Sciences, Dal′nauka, Vladivostok, 1996.
G. V. Alekseev and D. A. Tereshko, Inverse Extremum Problems for Stationary Equations of Heat Convection [in Russian], Preprint of Institute for Applied Mathematics, Far East Division of the Russian Academy of Sciences, Dal′nauka, Vladivostok, 1997.
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Paris, 1972; Springer, Berlin, 1976.
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.
P. Panagiotopoulos, Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions, Birkhäuser, Boston, 1985.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chebotarev, A.Y. Variational Inequalities for Navier--Stokes Type Operators and One-Sided Problems for Equations of Viscous Heat-Conducting Fluids. Mathematical Notes 70, 264–274 (2001). https://doi.org/10.1023/A:1010267111548
Issue Date:
DOI: https://doi.org/10.1023/A:1010267111548