Analysis Seminar

An outstanding question of the theory of the incompressible Navier-Stokes equations is whether solutions are unique in the Leray class, i.e., the weak solutions that dissipate energy. There is compelling numerical evidence due to Jia and Sverak of a reflection symmetry-breaking phenomenon leading to non-uniquness. In this talk we propose a new non-uniqueness scenario based on breaking of the (discrete) scaling symmetry, demonstrated in a shell model of the Navier-Stokes equations first formulated by Obukhov. We construct data in a critical space that gives rise to distinct Leray solutions which are approximately self-similar and ``smooth'' (in the sense of exponentially decaying energy spectrum) for positive times.