Estimates of the distance to the set of divergence free fields

S. Repin

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 44, Tome 425 (2014), pp. 99-116

Résumé

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.