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Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 1, Pages 37–66 (Mi izv1470)  

On a boundary value problem for the time-dependent Stokes system with general boundary conditions

I. Sh. Mogilevskii


Abstract: Solvability in Sobolev spaces $W_q^{2l,l}$ is proved and properties of solutions are investigated for the following initial boundary value problem:
\begin{gather*} \frac{\partial\bar{\mathbf u}}{\partial t}=\nabla^2\bar{\mathbf v}+\nabla p=\bar{\mathbf f},\qquad\nabla\cdot\bar{\mathbf v}=\rho\quadin\quad Q_T=\Omega\times(0,T),
\bar{\mathbf v}|_{t=0}=\bar v^0,\qquad B(x,t,\frac\partial{\partial x},\frac\partial{\partial t})(\bar{\mathbf v},p)|_{x\in\partial\Omega}=\bar{\mathbf\Phi}, \end{gather*}
where $\Omega$ is a bounded domain in $\mathbf R^3$ with smooth boundary, and $B$ is a matrix differential operator.
It is proved that under particular conditions imposed on the data of the problem and boundary operator $B$ there exists a solution $\bar{\mathbf v}\in W_q^{2l,l}(Q_T)$, $\nabla\rho\in W_q^{2l-2,l-1}(Q_T)$. The question of necessity of these conditions is investigated.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1987, 28:1, 37–66

Bibliographic databases:

UDC: 517.946
MSC: 35Q10, 76D05
Received: 09.06.1983

Citation: I. Sh. Mogilevskii, “On a boundary value problem for the time-dependent Stokes system with general boundary conditions”, Izv. Akad. Nauk SSSR Ser. Mat., 50:1 (1986), 37–66; Math. USSR-Izv., 28:1 (1987), 37–66

Citation in format AMSBIB
\Bibitem{Mog86}
\by I.~Sh.~Mogilevskii
\paper On a~boundary value problem for the time-dependent Stokes system with general boundary conditions
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1986
\vol 50
\issue 1
\pages 37--66
\mathnet{http://mi.mathnet.ru/izv1470}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=835565}
\zmath{https://zbmath.org/?q=an:0615.35077}
\transl
\jour Math. USSR-Izv.
\yr 1987
\vol 28
\issue 1
\pages 37--66
\crossref{https://doi.org/10.1070/IM1987v028n01ABEH000866}


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  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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