The famous Euler equations describe the flow of an ideal, incompressible fluid: a fluid with no viscosity or internal friction, that cannot be forced into a smaller volume. Almost all nonlinear fluid equations are derived from these. However, there are still many unknowns about Euler equations. Mathematicians have long suspected the existence of initial conditions that cause the equations to fail, but they cannot prove this.
But recently, a preprint submitted by Chinese-American mathematician Thomas Yizhao Hou and his former graduate student and Chinese mathematician Jiajie Chen proved that a particular version of the Euler equations does indeed sometimes fail. This proof marks a major breakthrough – although it does not completely solve the problem of the more general version of the equations, it brings people hope that a more general breakthrough can ultimately be achieved.
As early as 2013, Thomas Hou and Guo Luo, who now works at Hang Seng University of Hong Kong, put forward a hypothesis: Euler equations will lead to a singularity. They developed a computer simulation of a fluid in a cylinder whose top half swirled clockwise while its bottom half swirled counterclockwise.
In the movement of these two opposite currents, other complicated situations occur, such as the flow of water circulating up and down. At the point where they meet, the vorticity of the fluid (a hydrodynamic concept describing the rotation of the fluid) grows at an extremely fast rate, and it seems that it will “blow up” at any time.
However, their research at that time could only be said to be enlightening for the existence of singularities, and there was no real evidence. This is because it is impossible for a computer to calculate infinity. It can calculate an approximation that is very close to the singularity, but it is not the exact one.
In fact, when detected by more powerful computational methods, the obvious singularities have disappeared. For this reason, Charlie Fefferman, a mathematician at Princeton University, commented on past research on this matter, claiming that these problems are so delicate that the road is littered with the wreckage of previous simulations.
In 2022, Thomas Hou and his former graduate student Jiajie Chen successfully proved the existence of the nearby singularity. They first carefully analyzed the 2013 study and found that the approximate solution seemed to have a special structure. As time goes by, the solutions of these equations will show a so-called “self-similar pattern,” and its shape will later look like its previous one, but it will be rescaled in a specific way.
Therefore, they thought that there was no need to study the singularity itself. On the contrary, they could pay attention to an earlier time point to study it indirectly. By amplifying this part of the solution at the correct rate (which is determined by the self-similar structure of the solution), they can simulate what will happen later. What they need to do next is to prove that there is an exact solution near the singularity.
Thomas Hou, Charles Lee Powell Professor of Applied and Computational Mathematics at the California Institute of Technology, specializes in numerical analysis and mathematical analysis. Born in Guangdong Province, China in 1962, he studied at South China University of Technology as an undergraduate and obtained his bachelor’s degree in 1982. His doctoral career was completed at the University of California, Los Angeles. From 1989 to 1993, he taught at the Courant Institute of Mathematical Sciences at New York University. He has been teaching at the California Institute of Technology since 1993. In 2011, he was elected as a Fellow of the American Academy of Art and Sciences.
The other author of the study, Jiajie Chen, graduated from the School of Mathematical Sciences at China’s Peking University and is currently a mathematician at New York University. During his postgraduate years, he proved that various fluid equations can “blow up.”