Global wellposedness and local regularity for Navier–Stokes Equations

The question whether or not energy solutions to the Navier–Stokes equations are smooth is one of the seven Millennium problems. In the talk, an elementary introduction to the problem will be given and basic facts of the theory of those celebrated equations will be explained. Some general analytical questions will be raised in connection with investigations of differentiability properties of solutions to the Navier–Stokes equations. Another important aim of the talk will be to highlight the great contribution of members of the Laboratory of Mathematical Physics of LOMI/POMI (as a part of St. Petersburg's school of Mathematical Physics and Partial Differential Equations) to the theory of the Navier–Stokes equations starting with pioneering results of O. A. Ladyzhenskaya and her colleagues K. K. Golovkin, A. V. Ivanov, L. V. Kapitanskii, A. P. Oskolkov, and V. A. Sollonikov up to the latest works of the current members.