Weak solutions of the Euler and Navier-Stokes equations

Magnus Svärd
Applied and Computational Mathematics
Department of Mathematics
University of Bergen
Bergen, Norway

Abstract:

In an ideal world, an engineer could set up a CFD numerical simulation that to within a predefined tolerance approximates the true solution of the underlying PDE system. At present, this is impossible, even if we only consider the basic flow equations without any further modelling. It is not merely the accuracy that is impossible to predict a priori. We can not know if we have approximated any solution, let alone the true solution, on a sufficiently fine grid. We do not even know if there exists any solution for a general flow case.

In this talk, I will discuss weak solutions of the Euler and Navier-Stokes equations. Weak solutions are a first step towards a more complete well-posedness theory which in turn is a prerequisite for predictive CFD. Throughout the years, a plethora of schemes have been designed with various properties. I will focus on the non-linear stability properties entropy, kinetic energy and positivity, and their relations to weak solutions.