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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:
\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

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}