# Estimates of the distance to the set of divergence free fields and applications to analysis of incompressible viscous flow problems

## Prof. Dr. Sergey Repin

Feb. 15, 2018, 1:45 p.m. S2 416-1

We discuss mathematical questions that play a fundamental role in quantitative analysis of incompressible viscous fluids and other incompressible media.

Reliable verification of the quality of approximate solutions requires explicit and computable estimates of the distance to the corresponding generalized solution. In the context of this problem, one of the most essential questions is how to estimate the distance (measured in terms of the gradient norm) to the set of divergence free fields. It is closely related to the so--called inf--sup (LBB) condition or stability lemma for the Stokes Problem and requires estimates of the LBB constant. We discuss methods of getting computable bounds of the constant and respective estimates of the distance
to exact solutions of the Stokes, generalized Oseen, and Navier--Stokes problems.