University of Jyväskylä and Steklov Institute of Mathematics at St Petersburg

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 espective estimates of the distance to exact solutions of the Stokes, generalized Oseen, and Navier-Stokes problems.