The two-dimensional incompressible Euler equation is globally well-posed for smooth initial data — solutions exist for all time and remain unique. This is one of the cleanest results in fluid dynamics: unlike the three-dimensional case (where global regularity is a Millennium Prize problem), the 2D problem is solved. Solutions exist forever.
But existing forever doesn't mean staying smooth forever. Alazard and Said (arXiv:2603.13079) prove that for a dense set of initial conditions, solutions lose regularity in infinite time — they become progressively rougher, creating smaller and smaller scales of motion as time advances. This resolves a conjecture of Yudovich.
The result is about genericity, not pathology. It's not that some specially constructed initial data leads to roughening — it's that the smooth initial data that maintains its regularity forever is exceptional. The generic behavior is degradation. Starting from almost any smooth initial condition, the solution will eventually develop structure at arbitrarily fine scales.
The mechanism is the creation of small-scale vorticity gradients through the nonlinear stretching and folding of the velocity field. In 2D, vorticity is conserved along fluid trajectories (unlike 3D, where it can be amplified), but the gradients of vorticity are not conserved. Neighboring vorticity contours can be brought closer together, steepening the gradient without violating the conservation law. Over infinite time, this steepening creates arbitrarily fine structure — the solution remains well-defined but becomes rougher in any finite Sobolev regularity class.