Finite-time blow-up for Euler and Navier–Stokes equations

Determine whether smooth solutions to the incompressible Euler equations and to the incompressible Navier–Stokes equations admit finite-time singularities (blow-up) on a Riemannian manifold.

Background

The paper reviews universality phenomena in dynamical systems, with particular attention to Euler and Navier–Stokes flows. It recalls that the existence of finite-time blow-up for smooth solutions remains a central unresolved issue in PDEs and dynamics. This motivates Tao’s universality program and subsequent work embedding complex dynamics and computation into fluid flows.

The authors emphasize that computational universality might offer a route to constructing singular behaviors, yet the fundamental question of whether smooth Euler or Navier–Stokes solutions can blow up is still unresolved, and remains a benchmark open problem for the field.