Fluid dynamicist here. I think these sort of blow-up results get far too much attention from the popular science press, and I don't know why. This is practically a pure math problem. Both the Euler and (constant viscosity) Navier-Stokes equations are known to be flawed models in certain situations. And corrections for basically any identified flaw exist. Just because a certain category of flaw hasn't been proved yet doesn't mean much; its correction might already be published!
From what I understand, in the 1930s Jean Leray provided that the 2D (constant viscosity) Navier Stokes have smooth and unique solutions. And in the 1960s Olga Ladyzhenskaya proved that for certain non-constant viscosity laws (which are a lot more realistic than constant viscosity), the 3D Navier Stokes equations have smooth and unique solutions. From a physics perspective, there is no mystery here in terms of whether a properly posed problem will blow-up.
A lot of people (for example: [0]) link the question of whether the Navier-Stokes equations have smooth and unique solutions to turbulence, which makes little sense to me. Take the situations mentioned in the previous paragraph where proofs exist. The turbulence problem still exists there! The problem of turbulence is one of computational complexity. Turbulent flows seem to require a lot of computation.
Yeah, its kinda like how brownian motion has derivates of distributions which are not physical, and you can actually measure particles moving with super-highspeed cameras and they exhibit deterministic motion locally. So the I'd say the particles in both cases don't care too much about our mathematics.
Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question.
Turbulence indeed is high Kolmogorov complexity (w.r.t some reasonable grammar of numerical mathematics) and thus high computational complexity. But the question of N-S equation's adequacy comes before the question of computing.
I haven't worked on computational fluid dynamics in a long time -- happy to get insights from specialists.
> Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question.
No one I know working in turbulence regards the Navier-Stokes equations as a bad model for turbulence. They typically regard the equations as a source of truth about turbulence.
Focusing on turbulence in incompressible fluids [0], almost every time I've seen a comparison of "direct numerical simulations" using the Navier-Stokes equations against experiments measuring turbulence, the two matched well. Off the top of my head the only time I've encountered a discrepancy was deemed not due to the Navier-Stokes equations themselves but due to not modeling the initial or boundary conditions correctly [1]. Initial conditions refer to the flow field at the start of the simulation. Boundary conditions refer to conditions at a boundary, like a wall. Initial condition modeling is a big issue in certain flow instabilities like the Rayleigh-Taylor instability to my knowledge. The Navier-Stokes equations are chaotic, so they are sensitive to small changes in the initial conditions.
I'm not aware of anyone working on flow instabilities like the Rayleigh-Taylor instability who attributes a discrepancy between the experiments and simulations to an inadequacy of the Navier-Stokes equations. And to my knowledge I don't think any of the mathematicians working on the blow-up problem regard flow instabilities as clues indicating where to focus their attention to make the Euler/Navier-Stokes equations blow-up.
[0] You could, of course, pick a situation where the Navier-Stokes equations are known to not work well independent of turbulence considerations.
[1] Having worked a bit on initial and boundary condition modeling, though not in the context of the Rayleigh-Taylor instability, I can say that initial condition and boundary condition models are often bad, and they are understudied.
For high Knudsen numbers [0] the Navier-Stokes equations will be inaccurate. You can use a different approach based on the Boltzmann equation [1] like DSMC [2] for example.
As I wrote elsewhere there are good reasons to not use the Boltzmann equation everywhere [3]. Navier-Stokes is simpler and often completely adequate.
There are surely other situations as well; this is just what comes immediately to mind as I had a class covering some of this.
I'll believe that deep learning has really achieved human level performance when the algorithm looks at the Navier-Stokes equations and says, "eh, it's not like I'm ever actually going to use that after grad school".
Mathematics demigod Terry Tao came up with an extremely cool idea for how to prove blow-up results like this. https://www.youtube.com/watch?v=DgmuGqeRTto. The idea is to try and make a self-replicating "fluid computer" - think of our imaginary fluid as a continuous set of points which are all moving independently. If you set things up in very particular way, you can treat patches of fluid like components in a machine - these interact with each other, moving energy around to different parts of the fluid, until eventually you end up with a new copy of the machine - but smaller - and with more average velocity than the original. This then repeats, smaller and smaller, faster and faster, until within a finite time you have a singularity. He used this to prove blow up in some 'toy' versions of the Euler equations, with extra rules / parameters to make it work, but he does (or did) seem to think this has some chance of working in the real model.
(disclaimer - I do not have expertise in this field)
Great link. Hats off to the writers, as well as the researchers. Coming up with these ideas, implementing them, and them communicating them to a non-specialist audience in an engaging way takes serious effort. Thank you.
> In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity. [1]
> Quantum mechanical effects become significant for physics in the range of the de Broglie wavelength. For condensed matter, this is when the de Broglie wavelength of a particle is greater than the spacing between the particles in the lattice that comprises the matter. [...]
> The above temperature limit T has different meaning depending on the quantum statistics followed by each system, but generally refers to the point at which the system manifests quantum fluid properties. For a system of fermions, T is an estimation of the Fermi energy of the system, where processes important to phenomena such as superconductivity take place. For bosons, T gives an estimation of the Bose-Einstein condensation temperature.
> Classical fluids [1] are systems of particles which retain a definite volume, and are at sufficiently high temperatures (compared to their Fermi energy) that quantum effects can be neglected [...] Common liquids, e.g., liquid air, gasoline etc., are essentially mixtures of classical fluids. Electrolytes, molten salts, salts dissolved in water, are classical charged fluids. A classical fluid when cooled undergoes a freezing transition. On heating it undergoes an evaporation transition and becomes a classical gas that obeys Boltzmann statistics.
> Chaos theory is [...] focused on underlying patterns and deterministic laws highly sensitive to initial conditions in dynamical systems that were thought to have completely random states of disorder and irregularities. [1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization.
> If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?
... in dynamic, nonlinear - possibly adaptive - complex systems.
> In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain
If there is a ZeroDivisionError because a denominator is zero, is that actually a differentiable function?
-x**-1
-1/x
There's a symbolic result at or approaching that limit. IDK how exactly that applies to quantum-scale fluid effects at room temperature? Shouldn't there be a triality in terms of relativity, post-relativity QM with hair, and fluids converging or diverging to infinity?
> The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem. [...] Explanation using liquid flow [...]
The "derivative" at a certain point is defined as the limit of the ratio between function differences and argument differences.
In the context of derivatives, "exists" means that this limit is a finite number.
If the derivative function has an expression which is a fraction whose denominator becomes null at a point, the derivative may exist or not exist at that point.
If the corresponding numerator at that point is non-null (i.e. the derivative is infinite), then the derivative does not exist and the original function is not differentiable at that point.
If the numerator is also null, the derivative may exist or not at that point, depending on whether the limit of the derivative exists or not.
Are there no symbolic results from derivatives of any order?
Is it more correct to say that, if the denominator is zero at that point, the derivative is just non-Real because it's e.g. `n*x*oo`?
Practically, do we just say that such actually discontinuous functions are still mostly differentiable but the derivative does not exist in non-symbolic space?
> ... Quantum thermodynamics, fluids, and chaotic divergence
FWIW, does [quantum] thermodynamics predict emergent behaviors amongst self-organizing systems apparently at least temporarily contradicting a tendency to entropic decay?
My understanding is that no: fluid dynamics, quantum fluid dynamics, and quantum chemistry are not sufficient to describe and thus cannot predict emergent behaviors in complex nonlinear - possibly emergently adaptive - complex systems.
Perhaps before describing physical systems with current best known descriptions as multi-field (QFT,QQ,) wave-particle[-fluid] interactions with convergent and divergent e.g convection, it's appropriate to compare the difference between Classical and Quantum Wave Interference:
https://en.wikipedia.org/wiki/Wave_interference#Quantum_inte...
> some of the differences between classical wave interference and quantum interference: (a) In classical interference, two different waves interfere; In quantum interference, the wavefunction interferes with itself. (b) Classical interference is obtained simply by adding the displacements from equilibrium (or amplitudes) of the two waves; In quantum interference, the effect occurs for the probability function associated with the wavefunction and therefore the absolute value of the wavefunction squared. (c) The interference involves different types of mathematical functions: A classical wave is a real function representing the displacement from an equilibrium position; a quantum wavefunction is a complex function. A classical wave at any point can be positive or negative; the quantum probability function is non-negative.
Thus our best descriptions of emergent behavior in fluids (and chemicals and fields) must presumably be composed at least in part from quantum wave functions that e.g. Navier-Stokes also fit for; with a fitness function.
From what I understand, in the 1930s Jean Leray provided that the 2D (constant viscosity) Navier Stokes have smooth and unique solutions. And in the 1960s Olga Ladyzhenskaya proved that for certain non-constant viscosity laws (which are a lot more realistic than constant viscosity), the 3D Navier Stokes equations have smooth and unique solutions. From a physics perspective, there is no mystery here in terms of whether a properly posed problem will blow-up.
A lot of people (for example: [0]) link the question of whether the Navier-Stokes equations have smooth and unique solutions to turbulence, which makes little sense to me. Take the situations mentioned in the previous paragraph where proofs exist. The turbulence problem still exists there! The problem of turbulence is one of computational complexity. Turbulent flows seem to require a lot of computation.
[0] https://www.quantamagazine.org/the-trouble-with-turbulence-2...