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On a boundary value problem for the time-dependent Stokes system with general boundary conditions. (English. Russian original) Zbl 0615.35077

Math. USSR, Izv. 28, 37-66 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 1, 37-66 (1986).
The paper deals with the initial-boundary value problem \[ \partial \bar v/\partial t-\nabla^ 2 \bar v+\nabla p=\bar f,\quad \nabla \cdot \bar v=\rho, \] \(\bar v|_{t=0}=\bar v_ 0\), \(B(x,t,\partial /\partial x,\partial /\partial t)(\bar v,p)|_{x\in S}={\bar \phi}\), \(x\in \Omega \subset R^ 3\), \(S=\partial \Omega\), \(t\in (0,T).\)
The main result is the existence and uniqueness theorem for the solution (\=v,p) of the above problem. Further, various modifications of the boundary operator B are considered. Also necessary conditions for the existence of a solution are derived.
Reviewer: J.Bock

MSC:

35Q30 Navier-Stokes equations
35G30 Boundary value problems for nonlinear higher-order PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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