Mathematicians prove forced Navier-Stokes solutions are not unique in the Leray-Hopf class

IAS - March 2022
For almost 200 years, scientists and engineers have used the so-called Navier-Stokes equations to describe how fluids behave. These partial differential equations arose from Newton’s second law—that an object’s acceleration depends on its mass and the net force acting upon it—and incorporate internal forces, such as pressure, and external forces, such as gravity, on the movement of fluids. Among their applications, the Navier-Stokes equations, and their variations, are the main models used today for determining currents in the oceans, airflow around airplanes and cars, smoke dispersion from chimneys, and water flow in pipes.
But while the Navier-Stokes equations are useful for describing the physics of many important phenomena, at a theoretical level, exactly how accurately do they describe the behavior of fluids?
Mathematicians around the world have devoted countless hours to trying to answer that question. Recently, three mathematicians—all connected by their participation in a reading group at the Institute for Advanced Study (IAS) in Princeton, New Jersey—took a major step forward in answering it.
To do so, they focused on a set of solutions for Navier-Stokes in three dimensions called the Leray-Hopf solutions. The solutions, created in the early half of the 1900s, made it possible to view the Navier-Stokes equations “as a complete and self-consistent Newtonian model” that would “uniquely determine the future state of a system from complete information about its current state”—as long as uniqueness could be established, said Vladimír Šverák, a professor in the School of Mathematics at the University of Minnesota and current member at the IAS. Šverák explained that the solutions work under general assumptions that the initial velocity field is divergence-free and has finite energy.
While Leray worked with the unforced equation, Šverák noted, the solutions also exist if force is added. But are they unique? This is what the three mathematicians—Dallas Albritton, Elia Brué, and Maria Colombo—answered. Their proof revealed two distinct Leray solutions with zero initial velocity and identical body force, indicating that the exact same fluid starting from the exact same conditions could end up behaving in two distinctly different ways.
Based on this revelation, the Navier-Stokes solutions are not unique in the Leray-Hopf class in the presence of a force. The finding triggers a significant rethinking of the Navier-Stokes equations, building on earlier works by Hao Jia, Julien Guillod, and Šverák that produced striking numerical evidence in favor of nonuniqueness in the Leray class and by Tristan Buckmaster and Vlad Vicol that proved nonuniqueness of weak solutions of the Navier-Stokes equations, although not in the Leray-Hopf class.
"We're not saying that you can't predict the motion of a fluid, but mathematically, the Leray class, which is the class associated to the conservation of energy, is not the correct class," said Albritton, who is a current member at the IAS.
Reading group rewards
In addition to advancing understanding of the Navier-Stokes equations, the proof also shows the power of deep study and discussion, as the work that led to the proof grew out of a weekly, 7-person reading group at the IAS in which Albritton, Brué, and Colombo participated in 2020-2021. The reading group, organized by Camillo De Lellis, a faculty member at the IAS, focused on analyzing and understanding a two-part paper posted in 2018 on arxiv by Misha Vishik, a professor of mathematics at the University of Texas at Austin. The paper, which answers another famous open problem about uniqueness, proves nonuniqueness (part I) and instability (part II) in the Cauchy problem for the Euler equations of an ideal incompressible fluid.
Although the paper was believed to be important, it was not widely understood, said Albritton. The reading group motivated him and his colleagues to read the work carefully and consider it thoroughly, including how it fit with their individual areas of mathematical expertise.
Together, the reading group attendees "had a shared interest in understanding the detail," said Colombo, a professor at the École Polytechnique Fédérale de Lausanne in Switzerland and former member at the IAS.
This interest led them to identify a "deep piece of information in Vishik that was not somehow common knowledge," she said, and eventually led them to start discussing whether “one could push Vishik’s result to a nonuniqueness result for Navier-Stokes in two dimensions.” Then, she added, “we started to explore if we could somehow take advantage of the third dimension in order to push the example.”
In particular, Brué said, the Vishik paper got them thinking of the problem in terms of allowing the flexibility of a force. Brué, who is a current member at the IAS, noted that without force, it would have been very difficult to get the rigorous result that they achieved with their proof. The unforced problem remains open.
The reading group resulted not only in the proof by Albritton, Brué, and Colombo but also in the attendees gaining “different ideas on how to interpret and combine them with other ideas and move forward,” Colombo said. Multiple other papers that grew out of the reading group discussions are in progress, and a set of lecture notes has already been made available. Recently, Albritton and Brué gave a series of four lectures on these works as part of the Institute's Special Year on the h-principle and Flexibility in Geometry and Partial Differential Equations. Nonuniqueness is a core aspect of this year’s program at the IAS, which is co-organized by De Lellis and IAS Distinguished Visiting Professor László Székelyhidi, Jr.
Next steps for Navier-Stokes
The proof is particularly noteworthy because it is related to the classical Navier–Stokes existence and smoothness problem regarding whether smooth (infinitely differentiable) solutions always exist in three dimensions. Despite significant partial differential equations studies on three-dimensional fluids and turbulence, the problem remains open. The Clay Mathematics Institute has called it one of the most important problems in mathematics and offered $1 million for a solution or counterexample.
The Albritton, Brué, Colombo proof provides some fresh insight by revealing that a possible implication of the hypothetical formation of a singularity in the solutions (namely, that solutions are not unique after the singularity forms) should be taken seriously. “In a sense, it tells you that if you add an external force to the problem, which makes it a bit easier to tackle, then the evolution is not uniquely determined,” said Camillo De Lellis, the IAS faculty member who led the Vishik reading group.
"It also gives a new surprising take on these equations, because it really tells you that if there’s a singularity going on, then you shouldn't expect uniqueness anymore," he said, adding that "it makes you start thinking that there's really something more complicated going on in three dimensions."
Although Albritton, Brué, and Colombo have completed their proof, the impact of the Vishik reading group continues to shape their thinking about fluid dynamics and partial differential equations. Additional, related proofs are in the works, perhaps nudging mathematicians ever closer to understanding fluids.
"There remain many mysteries in fluids," Albritton said, “and in some way or another, they are all hiding in our most basic—and in some sense, only!—tool for understanding them: the Navier-Stokes equations."