Euler blowup: the collaboration behind the headlines

The Euler equations are the Navier–Stokes equations with the viscosity $\nu$ set to zero. The smoothing term $\nu \Delta u$ disappears, and with it the loss of energy: for a smooth solution,

so without a force the energy stays exactly constant. Nothing in the equation damps small-scale motion.

The vorticity $\omega = \nabla \times u$ measures how fast the fluid spins locally. Taking the curl of the equation gives

The term $(\omega \cdot \nabla) u$ is vortex stretching: a vortex tube that the flow stretches spins faster, like a skater pulling in their arms. In two dimensions this term is zero, the vorticity is only carried along, and smooth solutions exist for all time (Wolibner and Hölder, 1933). In three dimensions it can feed on itself.

Beale, Kato and Majda proved in 1984 that a smooth solution of the unforced equations can break down at a finite time $T^*$ only if

So a blowup proof has to show that the vorticity becomes infinite, and fast. A force helps the construction: it can steer the flow into a configuration that stretches itself ever faster. The difficulty is that the force must stay smooth up to the blowup time, so the singularity has to come from the nonlinear stretching, not from the force.

Blowup for the unforced three-dimensional Euler equations from smooth initial data in all of space is still open. Elgindi (2021) proved blowup for less regular data, of class $C^{1,\alpha}$, and Chen and Hou gave a computer-assisted proof of blowup for smooth data in a domain with a boundary. The Boussinesq equations, which add buoyancy, behave like three-dimensional axisymmetric Euler flow away from the axis, which is why results often come in pairs.