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To any fluid physicist or mathematician here: Does this move us closer to a general solution for Navier-Stokes?

kkylin 7y ago
If what you mean by "general solution for Navier-Stokes" is something like the Millennium Prize problem (proving general existence & uniqueness of solutions for NS), then no, at least not as far as I can see. This work is about a different problem than NS: they are mainly interested in the advection of passive tracers, e.g., a blob of ink, in a chaotically-evolving fluid field. To do this, they assume a number of things: their fluid equations are subjected to stochastic forcing, which makes available a bigger bag of mathematical tools than deterministic chaotic systems (like high Reynolds number NS without stochastic forcing). They also assume periodic boundary conditions, which enables explicit calculations using Fourier series representations. (Physical boundaries play an important role in many kinds of turbulent flows but make the mathematical analysis harder.) And, in 3d, a hyperviscosity term (4th derivatives in space instead of the usual 2nd derivatives associated with viscosity), which makes solutions more smooth. That being said, proving new theorems in this area is hard. I think it's very nice work.
- conformist 7y ago
  That's a nice summary! As far as I remember, a periodic domain is fine for one version of the Millennium problem.
  - kkylin 7y ago
    Thanks! (I had the benefit of sitting down with one of the authors and having some of this explained to me.) And thanks for the clarification about the Millennium Prize.
atrettel 7y ago
This result applies to turbulent flows. Not all flows are turbulent, and the Navier-Stokes equations should describe laminar, transitional, and turbulent flows (3 major regimes of fluid flow). Transitional flows are not yet fully turbulent and can behave differently than a fully turbulent flow behaves. My background is in fluid dynamics, so I can't speak directly about the mathematical difficulties in proving the existence and uniqueness of solutions of the Navier-Stokes equations. However, I can say that any general solution to the Navier-Stokes equations must apply to laminar, transitional, and turbulent flows, so finding a particular result for turbulent flows alone is insufficient. Any such result would have to transcend its regime in some way. Surely I think more proofs like this couldn't hurt, but I do not see any direct connection between this result and the Navier-Stokes existence/uniqueness problem (coming from a fluid dynamics background). Hacker News has already discussed this result previously in an earlier thread: https://news.ycombinator.com/item?id=21771684 https://news.ycombinator.com/item?id=21771684
- mar77i 7y ago
  Damn it, you beat me to bringing up laminar flow!
lgmargolin 7y ago
I think it is relevant to note this proof is based on the incompressible Navier--Stokes equations. Real fluids are compressible. From the point of view of kinetic theory, even the compressible Navier--Stokes equations are only a first-order approximation to the Boltzmann equation. Further, many authors (including myself) have questioned whether the Chapman--Enskog approximation is justifiable for high Reynolds number flows. Certainly it is known that NS does not correctly describe shock flows, which are also high Reynolds number flows. I would say these papers represent an interesting mathematical exercise whose relevance to the physics of turbulence remains to be assessed.

If what you mean by "general solution for Navier-Stokes" is something like the Millennium Prize problem (proving general existence & uniqueness of solutions for NS), then no, at least not as far as I can see. This work is about a different problem than NS: they are mainly interested in the advection of passive tracers, e.g., a blob of ink, in a chaotically-evolving fluid field. To do this, they assume a number of things: their fluid equations are subjected to stochastic forcing, which makes available a bigger bag of mathematical tools than deterministic chaotic systems (like high Reynolds number NS without stochastic forcing). They also assume periodic boundary conditions, which enables explicit calculations using Fourier series representations. (Physical boundaries play an important role in many kinds of turbulent flows but make the mathematical analysis harder.) And, in 3d, a hyperviscosity term (4th derivatives in space instead of the usual 2nd derivatives associated with viscosity), which makes solutions more smooth. That being said, proving new theorems in this area is hard. I think it's very nice work.

conformist 7y ago
That's a nice summary! As far as I remember, a periodic domain is fine for one version of the Millennium problem.
- kkylin 7y ago
  Thanks! (I had the benefit of sitting down with one of the authors and having some of this explained to me.) And thanks for the clarification about the Millennium Prize.

atrettel 7y ago

This result applies to turbulent flows. Not all flows are turbulent, and the Navier-Stokes equations should describe laminar, transitional, and turbulent flows (3 major regimes of fluid flow). Transitional flows are not yet fully turbulent and can behave differently than a fully turbulent flow behaves. My background is in fluid dynamics, so I can't speak directly about the mathematical difficulties in proving the existence and uniqueness of solutions of the Navier-Stokes equations. However, I can say that any general solution to the Navier-Stokes equations must apply to laminar, transitional, and turbulent flows, so finding a particular result for turbulent flows alone is insufficient. Any such result would have to transcend its regime in some way. Surely I think more proofs like this couldn't hurt, but I do not see any direct connection between this result and the Navier-Stokes existence/uniqueness problem (coming from a fluid dynamics background). Hacker News has already discussed this result previously in an earlier thread: https://news.ycombinator.com/item?id=21771684 https://news.ycombinator.com/item?id=21771684

mar77i 7y ago
Damn it, you beat me to bringing up laminar flow!

lgmargolin 7y ago

I think it is relevant to note this proof is based on the incompressible Navier--Stokes equations. Real fluids are compressible. From the point of view of kinetic theory, even the compressible Navier--Stokes equations are only a first-order approximation to the Boltzmann equation. Further, many authors (including myself) have questioned whether the Chapman--Enskog approximation is justifiable for high Reynolds number flows. Certainly it is known that NS does not correctly describe shock flows, which are also high Reynolds number flows. I would say these papers represent an interesting mathematical exercise whose relevance to the physics of turbulence remains to be assessed.