Estimates of the distance to a set of solenoidal fields and applications to a posteriori estimates for problems in the theory of viscous incompressible fluids

We discuss the question of estimating the distance between a given function $v\in W^{1,p}(\Omega,R^d)$, $p>1$ (with certain conditions on the boundary $\Gamma$) and a set of solenoidal (divergence-free) fields with the same boundary conditions in terms of the norm $\|\nabla v\|_{p,\Omega}$.
These estimates are important for many reasons, particularly because solutions to problems in hydrodynamics and electromagnetism are subject to the condition ${\rm div} u=0$, and the corresponding approximations satisfy it only with varying degrees of accuracy. A crucial role in deriving the estimates is played by the inf–sup condition (or the Ladyzhenskaya-Babushka-Brezzi LBB condition) and the constant $C_{LBB}$ arising in this condition. We discuss the issue of estimating this constant and methods for calculating it, as well as localized forms of the LBB condition, which contain a set of local constants associated with subdomains. They are used to estimate the distance between approximate and exact solutions to boundary value problems arising in the theory of viscous incompressible fluids.