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?



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?
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. 


