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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 practicall

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. [0] https://www.quantamagazine.org/the-trouble-with-turbulence-20190128/ https://www.quantamagazine.org/the-trouble-with-turbulence-2...

frozenport 5y ago
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.
andi999 5y ago
Thanks for the perspective. I am a bit puzzled why navier stokes (with constant viscosity) is one of clay millenium problems.
quanto 5y ago
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.
- btrettel 5y ago
  > 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.
  - andi999 5y ago
    Which situations are this in [0]?
    - btrettel 5y ago
      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. [0] https://en.wikipedia.org/wiki/Knudsen_number https://en.wikipedia.org/wiki/Knudsen_number [1] https://en.wikipedia.org/wiki/Boltzmann_equation https://en.wikipedia.org/wiki/Boltzmann_equation [2] https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo [3] https://news.ycombinator.com/item?id=31054992 https://news.ycombinator.com/item?id=31054992

frozenport 5y ago

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.

andi999 5y ago

Thanks for the perspective. I am a bit puzzled why navier stokes (with constant viscosity) is one of clay millenium problems.

quanto 5y ago

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.

btrettel 5y ago
> 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.
- andi999 5y ago
  Which situations are this in [0]?
  - btrettel 5y ago
    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. [0] https://en.wikipedia.org/wiki/Knudsen_number https://en.wikipedia.org/wiki/Knudsen_number [1] https://en.wikipedia.org/wiki/Boltzmann_equation https://en.wikipedia.org/wiki/Boltzmann_equation [2] https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo [3] https://news.ycombinator.com/item?id=31054992 https://news.ycombinator.com/item?id=31054992