Energetics of the Euler equation and self-similar blow-up
Roman Shvydkoy, University of Illinois-Chicago,
The existence of self-similar blow-up for the viscous incompressible fluids was a classical question settled in the seminal of works of Necas, et al and Tsai in the 90'. The corresponding scenario for the inviscid Euler equations has not received as much attention, yet it appears in many numerical simulations, most prominently in those based on vortex filament models of Kida's high symmetry flows. The case of a homogeneous self-similar profile is especially interesting due to its relevance to other theoretical questions such the Onsager conjecture or existence of Landau type solutions. In this talk we give an account of recent studies demonstrating that a self-similar blow-up is unsustainable the Euler system under various weak assumptions on the profile. We will talk about general energetics of the Euler system that, in part, is responsible for such exclusion results.