Image: Íñigo de Amescua/ICMAT

Euler’s equations, formulated in the 18th century by Leonhard Euler, describe the motion of ideal fluids, i.e., those that have no viscosity. These equations continue to pose some of the most profound and difficult problems in contemporary mathematics, especially with regard to the existence and uniqueness of their solutions.

In two dimensions, one of the fundamental results is Yudovich’s theorem, which guarantees that Euler’s equations admit a unique global solution when the initial vorticity (a quantity that measures the local rotation of the fluid) belongs to a specific class of functions, L^1∩L^∞. This theorem has served for decades as a reference for understanding when the behaviour of an ideal fluid is well determined by its initial conditions.

If these hypotheses are weakened (for example, if the vorticity is required to be in L^1∩L^p, with p a finite value), it is known that existence remains valid, but proving uniqueness in this regime remains an open problem of great interest.

In 2018, mathematician Misha Vishik took a decisive step forward by demonstrating that uniqueness can fail if an external force is introduced into the system, provided that this force belongs to the natural space L1(L1∩Lp). Their demonstration, developed in two articles totalling more than 130 pages, highlighted the extreme delicacy of the problem and opened up new lines of research into the instability of flows.

In a new paper, Ángel Castro (ICMAT-CSIC), Daniel Faraco (ICMAT-UAM), Francisco Mengual (Universidad de Sevilla) and Marcos Solera (University of Valencia) present an alternative and considerably simpler proof of this result, reducing the argument to about 30 pages. The main innovation of the article is the construction of a much simpler unstable vortex than the one originally proposed by Vishik. Unlike the previous vortex, the new example has zero mean and compact support, properties that greatly simplify the analysis and reinforce the conceptual clarity of the non-uniqueness mechanism.

Beyond the specific result for Euler’s equations, the ideas proposed in this work can be transferred to other fundamental models of fluid dynamics. In fact, in an independent study, the authors have demonstrated the non-uniqueness of solutions for the α-SQG (generalised Surface Quasi-Geostrophic) equation in the range 0<α≤1, within the critical functional class W^(s,p) with s<α+2p. The result includes the limiting case α=1, corresponding to the classical SQG equation, which is of great relevance in geophysics and meteorology.

Reference

Castro, Ángel, Faraco, Daniel, Mengual, Francisco and Solera, Marcos. “A proof of Vishik’s nonuniqueness theorem for the forced 2D Euler equation” Journal für die reine und angewandte Mathematik (Crelles Journal), vol. 2025, no. 824, 2025, pp. 253-288. https://doi.org/10.1515/crelle-2025-0025