The search for finite-time singularity solutions of the Euler equations for incompressible and inviscid fluids.

The search for finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by such that the flow contracts more rapidly into one direction than into the other, it can be shown that the dynamics of an axially symmetric flow with swirl may be approximated to a simpler hyperbolic system. By using the method of characteristics, it can be shown that generically the velocity flow exhibits multivalued solutions appearing on a rim at a finite distance from the axis of rotation, which displays a singular behavior in the radial derivatives of velocities. Moreover, the general solution shows a genuine blow-up, as a consequence of smooth initial data. This singularity is closely related to the singular solution found by T Elgindi in 2022 for a non-smooth initial data. These singularities are generic for a vast number of smooth finite-energy initial conditions and are characterized by a local singular behavior of velocity gradients and accelerations.