In the theory of partial differential equations (PDEs), an important problem is analyzing whether there is loss of regularity in finite time, even when there is smooth initial data. This is known as singularity formation, or blowups. A famous one is whether fluid dynamics represented by Navier-Stokes equations experiences such blowups, and is one of the unsolved Millennium prize problems.
A closely related problem is known as the Euler problem and it represents inviscid flow, i.e., it lacks the viscosity term present in the Navier-Stokes equation. Intuitively the lack of viscosity makes it easier for blowups to happen, but it is still an open problem if such blowups occur in free space (R^3).