Image: Íñigo de Amescua/ICMAT
A fundamental question in mathematical fluid theory is how and why classical solutions to equations describing flow motion can rapidly lose their regularity. The aim is to understand whether these equations behave correctly over time or whether, on the contrary, they develop irregularities or singularities. New work by Diego Córdoba (ICMAT-CSIC), Luis Martínez-Zoroa (Cunef University) and Wojciech S. Ożański (Florida State University) provides advances in this problem for the surface quasi-geostrophic equation (SQG).
The SQG equation describes the evolution of a temperature or density defined on a surface—for example, the ocean surface or an atmospheric interface—that determines the movement of the fluid. This equation is widely studied as a simplified analogue of the three-dimensional Euler equations. It retains enough structure to model realistic physical phenomena, while being more manageable than the full Euler equations. As a result, it has become a popular testing ground for studying ideas about instability, turbulence and singularity formation.
Previous research has shown that, in certain regimes, the SQG equation is ill-posed: very small perturbations of the initial state can cause drastic changes in the solution. Now, recent work demonstrates, for the first time in incompressible fluid mechanics, that the continuous loss of Sobolev regularity can occur in a controlled and explicit manner, even when the solutions remain unique and classically defined.
Mathematicians have managed to construct classical solutions that start out with a supercritical Sobolev norm arbitrarily small but immediately lose their regularity and continue to do so continuously and quantifiably as time passes. Instead of a sudden collapse at a single moment, the solution gradually becomes more irregular, losing regularity at a constant rate. This shows that even when solutions exist and are unique in a reasonable sense, their regularity can deteriorate endlessly. It is important to note that this loss of regularity occurs while the solution remains classical in time and space and remains compact.
This new result is entirely constructive, i.e., the authors explicitly construct solutions that exhibit this behaviour. Their construction is based on carefully arranged flow patterns that create strong stretching and compression effects, similar to what occurs near hyperbolic points in flows, while maintaining control over errors and interactions.
By assembling infinite structures of this type at different spatial scales, the authors produce a solution whose regularity continuously decreases over time. This multiscale mechanism is robust and accurate enough to track exactly how much smoothness is lost at each moment, with an exceptional level of detail.