The Navier–Stokes Equations: A never ending challenge?

More than 2500 years after the famous statement ????? ??? by Heracleitos the investigation of the mechanical and dynamical behavior of fluid flow is more than ever of fundamental importance. Due to a large number of technical, experimental and computational innovations and related theoretical problems the investigation of fluid flow represents a challenging and exciting subject requiring a wide variety of profound mathematical methods, efficient numerical algorithms and complex experimental simulations. Fascinating from the mathematical point of view, of course, is the fact that the fundamental equations of Navier–Stokes, formulated the first time by the French engineer Navier in 1822, could not be solved in the general three–dimensional case up to now. So the famous American Clay Mathematics Institute created the Navier–Stokes Millennium Price Problem and offered one Million US–Dollar for its solution, stating: „Although the Navier–Stokes equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory, which will unlock the secrets hidden in the Navier–Stokes equations“.

The lecture introduces the Navier–Stokes equations from an historical and physical point of view, touches some fundamental mathematical problems of viscous incompressible fluid flow and ends up with new regularity results.