# Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid?

This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold:

https://arxiv.org/abs/1709.10033

What's the current consensus of the community? Is the proof considered to be correct? Does this imply that the Navier-Stokes equations are not a valid model of fluid flow, or do we need a similar result for the smooth solutions of NS before we have to abandon/modify them?

• Welcome to MO! Note that as a general rule, asking about the correctness of a preprint is off-topic here: meta.mathoverflow.net/questions/927/… Your final question is a reasonable one though, and I see it already has a fine answer from Willie Wong.
– j.c.
Sep 12 '18 at 14:47
• @j.c. sorry I didn't know. However, the MO Meta thread seems to refer to crackpots: we can all agree that Buckmaster is a genius, not a crackpot. Sep 12 '18 at 18:07
• Name of this preprint: Buckmaster and Vicol - Nonuniqueness of weak solutions to the Navier–Stokes equation. Aug 1 '20 at 23:41

In regards to the question of the "consensus" or "correctness", I will only point out that Tristan Buckmaster has had a proven record of studying nonuniqueness problems for low-regularity solutions in incompressible fluids, and contributed significantly to the settling of Onsager's Conjecture on the nonuniqueness problem for incompressible Euler.

In regards to Navier-Stokes: weak solutions are called weak for a reason. To put it in simplest terms: the "solvability" of a PDE depends on what you accept as a valid solution.

(As a digression, this is not a problem unique to PDEs. Even in arithmetic if you work over $\mathbb{Q}$ the equation $x^2 = 2$ is not solvable, and if you work over $\mathbb{R}$ the equation $x^2 = -1$ is not solvable. Mathematics has a long history of "completing" the "space of admissible solutions" to solve previously unsolvable problems.)

There's an obvious trade off: if you enlarge the admissible solution space, you make it easier to solve an equation. But by making it easier to find a solution, you risk making it possible to find more than one solution.

(As an example, consider $x^3 = 3$. It is not solvable in $\mathbb{Q}$, it has a unique solution in $\mathbb{R}$, and it has three solutions in $\mathbb{C}$.)

In some sense you can think of existence and uniqueness as competing demands; a lot of PDE theory is built on figuring out how to restrict to a reasonable set of "admissible solutions" while guaranteeing both existence AND uniqueness.

In the context of Navier-Stokes, Leray (and Hopf) figured out a way to guarantee existence. People however have long suspected that their method does not guarantee uniqueness (in other words, that they are too generous when admitting something as a solution). Buckmaster and Vicol's work tries to carve away at this problem, by proving that for an even more generous notion of solution non-uniqueness can arise.

So no, we are absolutely nowhere near saying anything useful about physics or engineering; we are merely calibrating PDE theory.

As an aside, local existence and uniqueness for smooth solutions of NS hold. So a "similar result for smooth solutions" is in fact, impossible. This brings me back to the point of calibration:

• We know for sufficiently regular initial values, local-in-time existence and uniqueness of solutions to Navier-Stokes hold.
• We know that if we sufficiently relax the notion of solutions, global-in-time existence of solutions to Navier-Stokes hold.
• We know further that if an initial data admits a global weak solution that is in fact sufficiently regular, then that is the unique weak solution (in the sense of Leray-Hopf).

The main question on Navier-Stokes existence and uniqueness can be reformulated as: does there exist a sense of weak solution which guarantees global, unique solutions for all initial data, or is there a dichotomy where a sense of weak solutions that guarantees global solutions for all initial data is always too weak to guarantee uniqueness, and any sense of solutions guaranteeing uniqueness of solutions is always too strong to guarantee global solutions.

• "In regards to the question of the "consensus" or "correctness"": I understand now that discussing preprints is OT here, but just to clarify my point, I never doubted the competence of Buckmaster and Vicol (you'll see that Vlad too has a few astounding paper on mathematical fluid dynamics). It's just my understanding (but I'm not a math researcher, so I may be wrong) that usually mathematicians don't consider proofs in preprints correct until they have been published, unless the preprint has been around for a while and it has received favourable reviews. Since this is a preprint from 1/ Sep 12 '18 at 18:13
• 2/ famous mathematicians which has been around for a while, I thought that it had received some circulation among the community. This doesn't mean it's correct (Wiles' first proof had been around for a while, before the error was found), but I just want to make clear that my expectation was "yes, most experts think it's ok". I was not trying to cast doubts on the validity of the preprint. Concerning weak solutions: aren't weak solution useful for physics/engineering? Correct me if I'm wrong, but weak solutions admit shocks, and we do observe shocks in real fluid flows...of course shocks 2/ Sep 12 '18 at 18:18
• @DeltaIV: in regards to shocks, here are a few comments: (a) shocks form in compressible fluids, which is a completed different regime from the incompressible Euler/Navier-Stokes. (b) There are many different ways to understand "weak solution" (as already evidenced in my answer above). There are different levels of weakness. (c) Finally, the powerful application of weak solutions to understanding shock phenomenon is based on plane-symmetric flows. In the general case (which is still very poorly understood) one typically treats shocks as free boundary problems for classical solutions. Sep 12 '18 at 18:48
• (d) Shocks ARE useful in physics/engineering precisely because starting from smooth solution such singularities develop. If shocks were only to be an artifact of weak solutions (in analogy to the situation with your question) and were to not arise from smooth initial data, then physics and engineering would not care about it. Sep 12 '18 at 18:51
• (e) finally, from the physics point of view: one should remember that for the compressible euler equations, after the formation of shocks one also has an issue about uniqueness of weak solutions. The solution to this problem was not to throw out Euler as a good model for fluids, but to augment the notion of weak solution with a entropy condition forcing the solution to be unique. Compare this to what I wrote earlier about calibrating the notion of weak solutions. Sep 12 '18 at 18:58

To avoid people finding this question via google and wondering about the correctness of this paper, I want to point out that the paper has now been accepted by the Annals; see here.

• This is an interesting piece of extra information, but obviously it doesn't settle the question about correctness. Oct 24 '18 at 19:49
• @ChristianRemling: Of course not, but it's probably as good as we're going to get here on MO. Oct 24 '18 at 20:22