Estimates of the Distance to the Set of Divergence Free Fields and Applications to Quantitative Analysis of Incompressible Fluids

Sergey Repin

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Submission date: 06. May. 2014
Pages: 17
published in: Journal of mathematical sciences, 210 (2015) 6, p. 822-834 
DOI number (of the published article): 10.1007/s10958-015-2593-0
Bibtex
MSC-Numbers: 65F30, 35Q35, 65N15
Keywords and phrases: Incompressible fluids, inf--sup condition, distance to divergence free fields, Stokes, Oseen, and Navier-Stokes problems, computable bounds of the distance to the generaliz
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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). These estimates are connected with 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 cases. We consider 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 includes the LBB constant for a certain basic problem. In the last section, we apply these estimates to problems in the theory of viscous incompressible fluids and deduce fully computable bounds of the distance to generalized solutions.