The paper deals 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., the Stokes, Oseen, and Navier–Stokes problems). The distance is measured in terms of the L q norm of the gradient with q ∈ (1,+∞). For q = 2, these estimates follow from the so-called inf-sup condition (or the Aziz–Babuška–Ladyzhenskaya–Solonnikov inequality) and require sharp estimates of the respective constant, which are known only for a very limited amount of cases. A way to bypass this difficulty is suggested, and it is shown that for a wide class of domains (and different boundary conditions), the computable estimates of the distance to the set of divergence free fields 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. Bibliography: 26 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 425, 2014, pp. 99–116.
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Repin, S. Estimates of the Distance to the Set of Divergence Free Fields. J Math Sci 210, 822–834 (2015). https://doi.org/10.1007/s10958-015-2593-0
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DOI: https://doi.org/10.1007/s10958-015-2593-0