The Navier-Stokes Challenge
For nearly 200 years, the Navier-Stokes equations have been the foundational theory describing fluid behavior. However, mathematicians suspect that "blow-ups" or singularities, representing unphysical fluid movements, might exist within these equations. Discovering such a singularity, or proving its absence, is a Clay Millennium Prize problem offering a $1 million reward.
Stable vs. Unstable Singularities
Previous research has identified stable singularities in simplified fluid equations. These are less sensitive to initial conditions. In contrast, singularities in realistic fluid theories, such as Navier-Stokes, are hypothesized to be "unstable." These unstable singularities require extremely precise initial conditions and have been nearly impossible to find using traditional methods.
New Approach to Unstable Singularities
A group of mathematicians has developed a computational method to identify these elusive unstable glitches. Their research, detailed in a recent preprint, reexamined simpler fluid equations known to have stable singularities and successfully uncovered additional potential blow-up scenarios, including unstable ones. This marks the first discovery of a possible unstable singularity in a multi-dimensional fluid model.
Research Findings and Outlook
The team identified various unstable singularity candidates in several other fluid equations. While these are not the million-dollar Navier-Stokes singularities and still require rigorous proof, their success in uncovering potential unstable singularities in simpler models provides optimism. Charlie Fefferman, who formulated the Navier-Stokes challenge, noted that the idea of an unstable singularity no longer prevents its discovery.
Previous Singularity Research
Historically, mathematicians have simplified the Navier-Stokes problem, often studying the Euler equations, which model frictionless fluids. Computer simulations have been a common tool to approximate conditions leading to blow-ups. A notable achievement was Thomas Hou and Guo Luo's 2013 simulation of a spinning digital liquid, which hinted at a singularity. In 2022, Hou and Jiajie Chen rigorously proved the existence of this stable singularity.
Challenges of Unstable Singularity Detection
Identifying unstable singularities through traditional computer simulations is exceptionally difficult. The precise initial conditions required make it akin to balancing a pen perfectly on its tip. Even minute numerical errors introduced by computers can prevent the formation of such delicate blow-ups, as Tristan Buckmaster from New York University explained.