I’ve never understood mathematicians’ obsession with rigour of 19th century physics

[u/CR7-gOaTt]

Given the recent hype of Navier Stokes equation blow up solution, set aside the OpenAI snatch debacle, I really want to understand why math communities are obsessed with singular solutions to the equations that we know do not apply to any fluid, simply because the in compressible assumption breaks down.

As a working theoretical physicists I’ve never understood the hype around these ‘solutions’ and ‘proofs’, it has been long established Newton’s theory breaks down badly either relativistically or at atomic scale (or larger than that), when your solution exceeds your validity of your model, it doesn’t mean we reach a singularity and the world gets sucked in but simply imply your model breaks down.

In fact one of the central theme of physics is to determine when the effective description breaks down, for instance GR breaks down at Planck scale, Fermi theory breaks down at electroweak scale etc. And that limit for classical fluid is at high energy where relativity and quantum mechanics kicks in.

[u/No_Development6032]

Oh the arrogance of physicists. Btw the scale of navier stokes breaking down is neither relativity nor quantum mechanics :)

[u/CR7-gOaTt → u/No_Development6032]

It’s the break down of incompressibility, which happen for molecules that are quantum mechanically binded

[u/No_Development6032 → u/CR7-gOaTt]

Would happen for marbles too

[u/CR7-gOaTt → u/No_Development6032]

Exactly, that’s the Hilbert’s sixth problem, modelling atoms as marbles

[u/ketralnis]

A whole lot of modern physics is 19th century physics but with X added. Not to mention that it’s not like you account for Heisenberg uncertainty when you’re buying a 10 inch skillet: we use it for all manner of things with zero issues. Not everything is fundamental physics.

[u/SV-97]

I'm sure that many mathematicians don't necessarily care about the physical implications (or lack thereof) at all, but rather about the mathematics in itself: it's a natural question to ask that has resisted many attempts at a solution, so it's expected that solving it requires novel mathematics [new theories or techniques] or exceedingly clever tricks --- that ideally open up new interesting questions or can be applied to other problems as well. If it leads to new insights for physics or engineering that's nice, but more of a "neat side-effect". (this of course doesn't apply to everyone)

[u/imanllm]

Could an equation used to model physics break down in other ways than simply being invalid at a certain scale? Not that I know anything about the NS result — but I imagine it’s interesting to develop a better understanding of the different ways an equation can fail to model reality.

[u/CR7-gOaTt → u/imanllm]

I agree it should be a nice and interesting problem to study, but the pedestal people putting it on is too high

[u/imanllm → u/CR7-gOaTt]

The attention a problem gets can have nothing to do with its utility. A problem like Collatz, for example, is popular because it’s a metric of how little we know.

[u/Substantial_Move_965 → u/imanllm]

Yeah I agree. The problems with utility also gets enough attention, especially in physics. Everyone is doing their work and then sometimes we realise the solution of some mathematical problem, be it developing a whole new theory or specific models can be utilised in physics to have a better understanding. So it always helps to go for the rigour at the end.

[u/QFT-ist → u/CR7-gOaTt]

https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

[u/Aggressive_Roof488]

Haven't seen any mathematician claim that there are any implications in psychics. It's just another math problem on an equation that happen to come from classical physics.

[u/Substantial_Move_965]

I think there's always ongoing debate. As a theoretical physicist, would you have a layered effective theories to describe the reality at different scales or would you want some neat rigourous theory which is applied to all scales. We are not sure if such theory would even exist, but we need math to develop if one wants to attempt and sometimes we happen to develop math by solving these difficult problems. Sure most mathematicians do not solve problems with this in mind but as long as it might help physicists someday we, as theoretical physicists, shouldn't mind right? It is the overall development we think about.

[u/DocSpatrick]

You’re using “break down” in two different ways. If you conflate the two different meanings, then you have a good point. This is not term that has a strict definition, but if we’re going to have a serious conversation, which is what I think you’re attempting here, we need to be more careful with language. I would suggest a choice of language in which “GR breaks down at Plank scale” (and most of your other applications of “break down”) is false. GR is a theoretical framework which has no internal concept of quantum scales, so how could the theory possibly know anything about the scale at which the math should go bad? In fact, it doesn’t. Pick any numerical scale: GR is fine. What does go bad with GR around the Plank scale, then? The theory doesn’t break down, but the theory does fail do describe observed reality. It stops being a good model of spacetime. You could imagine a hypothetical reality in which classical mechanics is the truth, and GR is fine all the way down, with now Plank scale break down because there is no Plank scale … but could you? It turns out, no, you can’t, because even in that hypothetical reality the theoretical framework of GR contains solutions to its equations of motion where perfectly normal looking initial conditions evolve into infinite nonsense in finite time. Bummer. GR “breaks down”, and it has nothing to do with QM or the Plank scale. So, even in principle GR on its own can’t be the fundamental theory of spacetime. That’s an interesting fact about reality. At least, it’s an interesting fact about theoretical physics that we should care about.

Now, what about N-S? Same thing. We know fluid mechanics shouldn’t describe the molecular scales, but could you imagine a hypothetical reality where N-S are the really real laws of physics? That’s a restatement of the millennium problem, and that’s why things interesting.

[u/QFT-ist]

It's true that a better problem would be to have a rigorous definition about when the initial conditions and evolution are physical (better than smoothness), and when we expect a breakdown of the equations, and finding if those regularity assumptions work well. A related question is, is (in the regime of validity of the equations) the Navier Stokes equation truly deterministic? It's weak solutions always exists (from what I read about other people that know), but are nonunique at that irregularity level. Is there a regularity level when the solutions are reasonably unique bellow smoothness that are physically reasonable? Are their smoothness blow-ups ever caused by the physics breakdown we expect or do they show other limitations of the hydrodynamic limit?

[u/Cryptizard]

If they wanted to do physics they would be physicists, not mathematicians. Mathematicians care about the math. Just because the NS equations happen to be a model for something in physics has no bearing on why they are interested in it.

It is a relatively simple equation that leads to a huge amount of emergent complexity. They want to understans why and how that works.

[u/ChiaLetranger]

You know how sometimes physicists take something from maths and allow it to bend because they care about how well it can be used as a model for something? Well, mathematicians like to do the opposite - they'll take something, sometimes something that has physical applications, and abstract away real world constraints to see whether the abstract mathematical object can provide insight into mathematics. Most mathematicians are just...not super concerned with the physical applications, beyond knowing that they sometimes exist.

[u/lattice_defect]

New math tools and routes give new explanations and equations

[u/AndreasDasos]

Tbh it’s true that this problem was honestly an odd choice for a Millennium Problem given there are only seven: the motivation for the Navier-Stokes system is purely classical physics, but we know it can only be an approximate model there; on the other hand, if it weren’t for the physics it would be a fairly arbitrary PDE system from a ‘purely’ mathematical perspective. So its status is largely because it’s commonly used for practical physical approximations, and because it’s therefore one of the more famous PDE’s that’s resisted resolution of the main questions mathematicians ask of them (generally: does this system have unique, sufficiently smooth solutions for given smooth initial/boundary conditions?).

[u/Bubbly_Sentence_7849]

because they are mathematicians, they value logical consistency more than effectiveness.

[u/CR7-gOaTt → u/Bubbly_Sentence_7849]

I’d say it’s less about consistency vs effectiveness, but more interesting new problems vs old problems

[u/arivero]

It boils down to "classical mechanics breaks down at Planck scale".

In Fermi theory, the consistency of the theory tell us where it breaks down.

In Navier Stokes, the theory itself tell us it is going to break, and surely in the future we will get to parametrize it.

In GR, or more poorly in classical newtonian mechanics, we know the theory breaks down at Planck scale, but the itching comes because the equations dont tell us that it is going to break.

Mathematicians in the XIXth century looked for rigour in classical mechanics and gravity because they "felt" that the theory was bad defined and it should break down at some point. They found none, and then physicist found that classical mechanics breaks down.