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.

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.