After the OpenAI-derived counterexample to the Navier-Stokes existence and smoothness problem1, I thought I would try to put into writing my own interpretation of the result. I'm not a mathematician, I'm a physicist and meteorologist, so I have a good grasp of fluid mechanics, but from the physical sciences and numerical modelling side.
If the OpenAI result is verified, we have at least one example of a combination of smooth initial conditions and smooth forcing that leads to finite time blow-up (i.e. infinite velocities). To get finite-time blow-up there are two knobs to turn, the initial velocity field [u0(x)] and the external forcing [f(x,t)]2.
According to the blog post by OpenAI announcing the result, they started with u0=0 everywhere and created a forcing that lead to finite-time blow-up. The blog post describes it as follows:
The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.3
So my interpretation stands as: The OpenAI researchers were able to find/define a smooth f(x,t) that drives the fluid to velocity divergence in finite time. According to some other things I've read it's based on ideas from a few years ago where the forcing is in the form of precise small-scale high-frequency perturbations4. Apparently, part of the difficulty is forcing the velocity field to blow-up while keeping the forcing smooth as described by Tao5. I think that's what the slightly unclear last sentence alludes to.
Be that as it may, I wanted to tie my interpretation to certain statements that are often made regarding the value of the Navier-Stokes existence and smoothness problem, and the relationship of these equations to real world fluids. An example of this type of statement is given in the write-up by Science6:
So, what did OpenAI researchers do—and did they prove the Navier-Stokes equations wrong? Effectively, they proved the Navier-Stokes equations sometimes run amok and produce an unphysical solution. They found a particular solution to the equations that, in a finite amount of time, evolves into a whorl in which the fluid moves at infinite speed. Nothing can move infinitely fast, not even light. So, the researchers essentially proved by example that there are situations in which the Navier-Stokes equations produce a nonsensical result.
Or this article from Scientific American7
The mathematical problem concerns the Navier-Stokes equations, which govern the flow of fluids—from the whirlpool circling your bathtub’s drain to the turbulent winds El Niño hurls across the North American continent. After two centuries of study, mathematicians haven’t been able to figure out whether these equations perfectly describe reality in every situation or if they sometimes admit aberrant mathematical blips that could never happen in a real fluid.
Real fluids are made of discrete molecules moving around under the influence of (mostly) electromagnetic and gravitational forces and bouncing off each other and any boundaries. Each molecule follows the rules of classical mechanics8. The Navier-Stokes equation emerges as the continuum limit of statistical mechanics for liquids and gases, meaning that the fluid is considered to consist of infinitely divisible and continuous matter. This works really well for things like water and air and many other things, but it breaks down when gases become too thin for example.
The way these popular science articles write about it, it sounds like there is some direct connection between the (solutions to the) Navier-Stokes equations considered as mathematical objects and the physics of real-world fluids.
Under normal circumstances we can say that the physics of fluids are governed by the Navier-Stokes equations and that the physical interpretation of a solution to the Navier-Stokes equations is a velocity field representing the flow of a fluid. But there are several different forms of the NS equations9 and the version defined by the Millennium Prize has a couple of simplifying assumptions compared to the most general form:
It concerns the case of incompressible flow, meaning that density does not vary with pressure. This is a good approximation for fluids like water, but often not for gases. And no fluid is perfectly incompressible.
It assumes a homogeneous fluid with constant viscosity (stickyness, friction).
It assumes that the second viscosity is zero (which follows from the incompressibility condition).
Thermodynamics is neglected.
Specifically for the millennium Prize, it assumes an infinite fluid in all three directions.10
None of the above are strictly true for general fluids, though they are often good approximations under normal circumstances depending on the problem and fluid under consideration.
When journalists and others write things like the quotes above, they imply that the counterexample has shown that the NS equations are not a complete description of real fluids, but I think that's the wrong interpretation.
Consider trying to recreate the result in the rea world, using for example water. You've built a giant tank of water and found a way of stirring the fluid in the right way using sound or lasers or something. As the velocity of the fluid increases your simplifying assumptions would break down long before you come anywhere close to blow-up.
As you approach the speed of sound in the fluid, the incompressibility criterion breaks down, temperature effects become important and the viscosity starts to depend on the flow state. If you were able to overcome these problems with, e.g. a hypothetical fluid with extremely high speed of sound you'd eventually get into a a regime where relativistic effects become important and the entire thing breaks down11.
The NS equations, as derived from statistical mechanics are still a perfectly valid description of the behaviour of real fluids as long as you obey the limits and assumptions made in the derivation.
Where does this leave the new result and its physical interpretation? I think the right way is like this: If you were able to create a perfectly homogeneous, perfectly continuous, perfectly incompressible fluid with constant viscosity, infinite heat capacity and infinite extent; in a world where relativity and quantum mechanics are not needed: Then, the Navier-Stokes equations would still not be the correct mathematical description for that fluid under all circumstances.
Does this mean that we need a new theory for fluid flow? I don't think so as the real world is protected from these mathematical blips by the breakdown of the necessary assumptions required for NS to be applicable in the first place.
Remember that from the perspective of molecules, fluids do not exist.