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 Zap. Nauchn. Sem. POMI, 2014, Volume 425, Pages 99–116 (Mi znsl6023)

Estimates of the distance to the set of divergence free fields

S. Repinab

a St. Petersburg Department of the Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023, St. Petersburg, Russia
b St. Petersburg State Polytechnical University, Polytechnicheskaya 29, St. Petersburg, Russia

Abstract: We are concerned with computable estimates of the distance to the set of divergence free fields, which are necessary for quantitative analysis of mathematical models of incompressible media (e.g., Stokes, Oseen, and Navier–Stokes problems). The distance is measured in terms of $L^q$ norm of the gradient with $q\in(1,+\infty)$. For $q=2$, these estimates follow from the so-called inf-sup condition (or Aziz–Babuška–Ladyzhenskaya–Solonnikov inequality) and require sharp estimates of the respective constant, which are known only for a very limited amount of cases. We suggest a way to bypass this difficulty and show that for a vide class of domains (and different boundary conditions) computable estimates of the distance to the set of divergence free field can be presented in the form, which uses inf-sup constants for certain basic problems. In the last section, these estimates are applied to problems in the theory of viscous incompressible fluids. They generate fully computable bounds of the distance to generalized solutions of the problems considered.

Key words and phrases: incompressible viscous fluids, inf-sup condition, distance to divergence free fields, Stokes, Oseen, and Navier–Stokes problems, computable bounds of the distance to the generalized solution.

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English version:
Journal of Mathematical Sciences (New York), 2015, 210:6, 822–834

UDC: 517
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Citation: S. Repin, “Estimates of the distance to the set of divergence free fields”, Boundary-value problems of mathematical physics and related problems of function theory. Part 44, Zap. Nauchn. Sem. POMI, 425, POMI, St. Petersburg, 2014, 99–116; J. Math. Sci. (N. Y.), 210:6 (2015), 822–834

Citation in format AMSBIB
\Bibitem{Rep14} \by S.~Repin \paper Estimates of the distance to the set of divergence free fields \inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~44 \serial Zap. Nauchn. Sem. POMI \yr 2014 \vol 425 \pages 99--116 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6023} \transl \jour J. Math. Sci. (N. Y.) \yr 2015 \vol 210 \issue 6 \pages 822--834 \crossref{https://doi.org/10.1007/s10958-015-2593-0} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84944705856} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Repin S., “Estimates of the Distance To the Set of Solenoidal Vector Fields and Applications To a Posteriori Error Control”, Comput. Methods Appl. Math., 15:4 (2015), 515–530
2. J. Math. Sci. (N. Y.), 224:3 (2017), 456–467
3. S. Repin, “On projectors to subspaces of vector valued functions subject to conditions of the divergence free type”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 46, Zap. nauchn. sem. POMI, 459, POMI, SPb., 2017, 83–103
4. S. Repin, “Localized forms of the LBB condition and a posteriori estimates for incompressible media problems”, Math. Comput. Simul., 145 (2018), 156–170
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