Ladyzhenskaya Boundary Value Problems Of Mathematical Physics Pdf

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This article is cited in 39 scientific papers (total in 41 papers)

Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations

O. A. Ladyzhenskaya, V. A. Solonnikov

Abstract: We consider the problem of finding the restrictions on the domain

$\Omega\subset R^n$, $n=2,3$, under which the space $\overset{\hat\circ}J{}^1_2(\Omega)$ of the solenoidal vector fields from $\overset{\circ}W{}^1_2(\Omega)$ coincides with the space $\overset{\circ}J{}^1_2(\Omega)$, the closure in $W_2^1(\Omega)$ of the set of all solenoidal vectors from $\dot C^\infty(\Omega)$. We give domains $\Omega\subset R^n$, for which the factor space $\overset{\hat\circ}J{}^1_2(\Omega)/\overset{\circ}J{}^1_2(\Omega)$ has a finite nonzero dimension. A similar problem is considered for the spaces of solenoidal vectors with a finite Dirichlet integral. Based on this, one compares two generalized formulations of boundary-value problems for the Stokes and Navier–Stokes systems. The following auxiliary problems are studied: 1)$\operatorname{div}\vec{u}=\varphi$, $\vec{u}|_{\partial\Omega}=0$; 2)$\operatorname{div}\vec{u}=0$, $\vec{u}|_{\partial\Omega}=\vec{\alpha}$; 3) $\operatorname{grad}p=\sum\limits^n_{k=1}\dfrac{\partial\vec{R}_k}{\partial x_k}+\vec{f}$.

Full-text article is available Full text: PDF file (1480 kB)

English version:

Journal of Soviet Mathematics, 1978, 10:2, 257–286

Bibliographic databases:

UDC: 517.994

Citation: O. A. Ladyzhenskaya, V. A. Solonnikov, "Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations", Boundary-value problems of mathematical physics and related problems of function theory. Part 9, Zap. Nauchn. Sem. LOMI, 59, "Nauka", Leningrad. Otdel., Leningrad, 1976,

81–116; J. Soviet Math., 10:2 (1978), 257–286

Citation in format AMSBIB

\Bibitem{LadSol76} \by O.~A.~Ladyzhenskaya, V.~A.~Solonnikov \paper Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier--Stokes equations \inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~9 \serial Zap. Nauchn. Sem. LOMI \yr 1976 \vol 59 \pages 81--116 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl2086} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=467031} \zmath{https://zbmath.org/?q=an:0388.35061|0346.35084} \transl \jour J. Soviet Math. \yr 1978 \vol 10 \issue 2 \pages 257--286 \crossref{https://doi.org/10.1007/BF01566606}

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Related articles on Google Scholar: Russian articles, English articles

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