Image: Álvaro Minguito/ICMAT

Euler’s equations, formulated in the 18th century, describe the motion of incompressible fluids (i.e., fluids whose volume is constant, such as water) without viscosity and constitute one of the pillars of fluid mechanics. Their solutions offer, based on initial conditions, a prediction of how the fluid will behave in the future.  So-called weak solutions describe highly irregular flows, in which the velocity of the fluid may cease to be smooth or even exhibit discontinuities. These solutions are very flexible and are particularly useful for studying turbulent phenomena, but they may not be unique and may exhibit behaviour that is very different from that expected from classical physics.

A new paper by Alberto Enciso (ICMAT-CSIC), Javier Peñafiel (ICMAT-CSIC) and Daniel Peralta-Salas (ICMAT-CSIC) shows the extent to which weak solutions of Euler’s equations in three dimensions can be extraordinarily flexible. The authors demonstrate that any classical and smooth solution, defined only in a limited region of space and time, can be extended to the entire space as a weak solution during that same time interval. This extension coincides exactly with the original solution where both are defined and retains a certain degree of regularity, although always below a critical threshold. The result is based on the convex integration technique, a tool that has revolutionised the study of fluid equations over the last decade.

Based on this extension theorem, the paper presents two particularly striking applications. The first demonstrates that, starting from a very simple initial configuration known as a vortex sheet, it is possible to construct an infinite number of different weak solutions that coincide with the prescribed sheet outside a small turbulent region. This phenomenon highlights an extreme form of non-uniqueness: the same initial state can give rise to an infinite number of mathematically valid evolutions.

The second application shows that even when starting from a completely smooth flow defined in a periodic domain, it is possible to modify it within a specific open region to obtain a weak solution that develops singularities in a set of space-time of positive measure. In other words, highly irregular behaviour can emerge in a localised manner, while the flow remains smooth and well-behaved outside the chosen region.

Reference: Enciso A, Peñafiel-Tomás J, Peralta-Salas D. An extension theorem for weak solutions of the 3d incompressible Euler equations and applications to singular flows. Forum of Mathematics, Pi. 2025;13:e21. doi:10.1017/fmp.2025.10012