Abstract
In a bounded domain of the space Rn (n=2 or 3) we consider the initialand boundary-value problem regarding the determination of the velocity vector
of the fluid, the pressure, and the density from the system of Navier-Stokes equation
and the continuity equations, as well as from the initial conditions for the velocity and from the adherence boundary conditions. It is proved that the three-dimensional problem is uniquely solvable on some finite time interval and, in the case of a small initial velocity vector and a small volume force, also on an infinite interval; however, the two-dimensional problem is uniquely solvable for all t⩾0 without any smallness restrictions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
S. N. Antontsev and A. V. Kazhikhov, The Mathematical Problems of the Dynamics of Non-homogeneous Fluids [in Russian], Novosibirsk (1973).
O. A. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1969).
O. A. Ladyzhenskaya (Ladyzenskaya), V. A. Solonnikov, and N. N. Ural'tseva (Ural'ceva), Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence (1968).
Sh. Sakhaev and V. A. Solonnikov, “Estimates of the solutions of a boundary-value problem of magnetohydrodynamics,” Tr. Mat. Inst., Akad. Nauk SSSR (1975).
S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Comm. Pure Appl. Math.,12, 623–727 (1959).
V. A. Solonnikov, “On a priori estimates for certain boundary-value problems,” Dokl. Akad. Nauk SSSR,138, 781–784 (1961).
V. A. Solonnikov, “Estimates of the solutions of nonstationary Navier-Stokes systems,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., Akad. Nauk SSSR,38, 152–231 (1973).
V. A. Solonnikov, “Estimates of the solutions of the nonstationary linearized Navier-Stokes equations,” Tr. Mat. Inst. Akad. Nauk SSSR,70, 213–317 (1964).
V. P. Il'in, “On the ‘embedding’ theorems,” Tr. Mat. Inst., Akad. Nauk SSSR,53, 359–386 (1959).
P. E. Sobolevskii, “Coercivity inequalities for abstract parabolic equations,” Dokl. Akad. Nauk SSSR,157, 52–55 (1964).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 52–109, 1975.
Rights and permissions
About this article
Cite this article
Ladyzhenskaya, O.A., Solonnikov, V.A. Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids. J Math Sci 9, 697–749 (1978). https://doi.org/10.1007/BF01085325
Issue Date:
DOI: https://doi.org/10.1007/BF01085325