7 ms·

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initia

The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. The incompressible Euler equations model a very strange and unphysical kind of fluid. Incompressibility means that the speed of wave propagation in such a fluid is infinite, which means that normal causality is not respected. Effects in such a fluid happen simultaneously with their causes. For example, if you apply a force to one end of a pipe full of Euler fluid, the fluid instantly starts coming out of the other end of the pipe, with no time taken for this effect to propagate from one end of the pipe to the other. You could use a long pipe full of Euler fluid as a superluminal communication device! Intuitively, it seems reasonable that in such an unphysical fluid, it would be possible to form a singularity even from smooth initial conditions. The difficulty, of course, is proving that intuition, which is what the paper is trying to do. 1) https://arxiv.org/pdf/2210.07191.pdf https://arxiv.org/pdf/2210.07191.pdf "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data", Jiajie Chen and Thomas Y. Hou.

nuclearnice1 4y ago
What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.
- isoprophlex 4y ago
  I think (not a physicist), simply put, an infinity or NaN value. As these are step-wise methods, having such values show up anywhere will seriously mess up subsequent calculation steps.
- prolyxis 4y ago
  A simple example of a function with a singularity is f(t)=1/t. Note that at t=0, f(t) is undefined due to division by zero. On either side of zero, the absolute value of f(t) approaches infinity. In this case, we are tracking the flow of an incompressible fluid over time. This flow is represented by a velocity field evolving over time, under the constraint of no net inflow/outflow of material into any region of space. Thus, the singularity corresponds to a portion of fluid speeding up and approaching an infinite speed as you approach some finite time. Because the fluid cannot be compressed, the only way the singularity can be produced is for a portion of the liquid to swirl, increasingly rapidly, about some point: hence the discussion in the article about vorticity. As isoprophlex pointed out, this undefined value of the velocity field prevents you from (or at least complicates) computing the further evolution of the fluid.
  - nico 4y ago
    Thank you for the great explanation. Do these swirls shed energy? Is it considered in these equations that for example friction within the swirls would slow them down (and hence not reach a singularity)?
    - pas 4y ago
      In real fluids yes, absolutely, they basically transform/branch/divide/split into smaller and smaller scale vortices and then those dissipate the energy into the fluid (heating it up a bit).
- wwalker3 4y ago
  The incompressible Euler equations model a fluid as a two-valued field. This means that at every point in space, the field has two values, density and velocity (1). To me (2), a singularity in a field like this means that one or more of the field values "blows up", i.e. goes to infinity as you run the time variable forward. But how could this ever happen? The Euler equations model the "conservation" (i.e. constant-ness) of three real physical quantities: mass, momentum, and energy. If these three quantities are finite and constant when you add them up over the whole field, how can any part of it "blow up" into an infinite value? The answer is that the blow-up must occupy a volume that shrinks as the blow-up grows, so the conserved quantities are still constant. The singularity would be infinitely small in space, and have an infinite value of density or velocity (or both). The hard question is, are these blow-ups merely artifacts of a particular numerical simulation technique, or are they essential somehow to the incompressible Euler equations themselves? That's what these papers are trying to figure out. To me, an "essential" (i.e. inherent-in-the-equations) blow-up seems intuitively reasonable because of the acausal nature of the field. When you simulate the incompressible Euler equations, it superficially looks like it's a physical fluid doing physical-fluid things, swirling and flowing around. But in a real fluid, a change in one part of the fluid propagates to the other parts at finite velocity, creating real cause and effect. An Euler fluid's time evolution is not a phenomenon that ripples forward through time in a normal way. Instead, every point in the fluid responds to every other point simultaneously. If you poke a cube of incompressible Euler fluid with your finger, there is no pressure wave that ripples through it, where the fluid parcels push each other along and get out of each other's way. Instead, the whole cube of fluid somehow instantly adopts a new flow pattern that conserves mass/momentum/energy in response to that finger-poke. 1) Note that velocity is a vector, since it has a direction. This means that in 2D the velocity is two numbers, and in 3D it's three numbers. So technically the 3D incompressible Euler equations have four values at every point: one density, and three velocity components, one each in the x, y, and z directions. 2) I'm a numerical simulation guy, not a mathematician. Real math experts have rigorous definitions of a singularity, e.g. in https://arxiv.org/pdf/2203.17221.pdf https://arxiv.org/pdf/2203.17221.pdf "Singularity formation in the incompressible Euler equation in finite and infinite time," Theodore D. Drivas and Tarek M. Elgindi.
  - vba616 4y ago
- NotYourLawyer 4y ago
  It’s when some physical quantity of the simulation becomes infinite. Pressure, particle velocity, etc.
  - TEP_Kim_Il_Sung 4y ago
    That usually indicates a phase change, with a separate set of applicable equations.
pencilguin 4y ago
I guess this answers why you can't just try it with liquid helium: even that isn't ideal enough.
WalterBright 4y ago
Best comment I ever read on HN.
- wwalker3 4y ago
  Wow, I'm honored :) These days, I try to only comment when an article is really in my wheelhouse, but that's not very often, given my narrow interests in fluid dynamics and computational physics.
  - WalterBright 4y ago
    > I try to only comment when an article is really in my wheelhouse Which is why your comment is exceptionally worthwhile. I also know enough about fluid mechanics to both understand and appreciate it. People often wonder on HN what the point of a STEM degree is (after making money). To me I've had a lifetime of pleasure from understanding how things work. It's so much better than things being mysterious black boxes. I once asked a date if she wanted to understand how airplanes worked. She said no, that understanding them would make her afraid of flying. For me, it was the opposite. Knowing how the airplanes fly and how it all works made me a much less anxious passenger.
graycat 4y ago
Yes, in pure/applied math, we know a lot about various cases of approximation. But in practice there are more cases of approximation, and, right, the Euler equations are another such case. Or, to be a little flippant, generally in applications to real problems, we look at a lot of the features and throw out some, modify some, and actually honor some!! So, a question is, can we improve our ability to make such approximations and know something about the accuracy of the solutions we will get? E.g., for the Euler equations, will that approximation of an "incompressible" fluid ever work in practice and, if so, when and, there, how accurate can/will it be? Or, what about, hmm, just to be picky and pick something, friction on the side of the tube? What if the tube is not a perfect tube? A few grains of dirt: What if the liquid is water but, like most real water, has some solids floating around in it? Right, we can say, so there are a few grains of dirt floating around in the water, and they won't matter -- to be picky, that's an approximation, and we are likely correct, but where is an actual math theorem that says we are correct or how correct, i.e., accurate, are we? Right, a few grains of dirt -- we don't much care. But that's practical judgment and not really theorem/proof math. And similarly for other approximations we get as we throw out, modify, or honor real features? So, as stated, this is too difficult as a pure/applied math research direction. Okay, ..., then, is there anything at all in that direction that might be not absurdly difficult as a research direction? Or, to be simplistic, we work hard and get a numerical solution to a boundary value problem. Now someone tweaks the boundary. Can we say that our numerical solution is only tweaked? Or, when can we say that small changes in the problem statement will result in only small changes in the solution? Right, we are into some topology and looking for a case of continuity .... Hmm .... If we had some linearity ...!!! Right, the two pillars of analysis are continuity and linearity ...! But here with Euler we were considering nonlinear partial differential equations! Again I ask, is there any hope we can do anything for some corresponding math??
civilized 4y ago
This raises a question I hadn't thought of before. Real-world fluid flow is ultimately well-modeled by the equations of many-body Newtonian mechanics, right (atoms bumping around)? Are those equations vulnerable to blow-ups?
- vba616 4y ago
  This makes me think of: https://en.wikipedia.org/wiki/Sonoluminescence https://en.wikipedia.org/wiki/Sonoluminescence I would think that nothing in reality is infinite, but allegedly sound waves collapsing bubbles in a fluid can cause a very small amount of plasma to become hotter than the sun and emit light. Some controversial research claims it might even be possible to create atomic fusion this way.
- wwalker3 4y ago
  Pretty much any mathematical model of a real phenomenon can have some sort of singularity or discontinuity in it. If you model atoms as dimensionless points (1), then any kind of force law with the distance between atoms in the denominator can lead to a singularity when that distance is zero. In practice, you write the simulator to disallow this, but it's still there in the equations, you're just ignoring it. If you model your atoms as finite-sized but incompressible billiard balls, then when they hit each other it's a discontinuity, since they instantly change direction when they collide. These collisions conserve total momentum and energy, but they're unphysical because real physical quantities can't jump from one value to another (in classical physics). Even if you model your atoms as little rubber balls, the model can still be singular. Linear elasticity (the most common choice) allows you to compress a finite-sized object down to zero size with finite energy, which yields infinite energy density. Again, you'd have to disallow that in the simulator, which is very practical, but not theoretically satisfying. 1) https://en.wikipedia.org/wiki/Molecular_dynamics https://en.wikipedia.org/wiki/Molecular_dynamics is the typical method of atomistic simulation. 2) https://en.wikipedia.org/wiki/Linear_elasticity https://en.wikipedia.org/wiki/Linear_elasticity
  - civilized 4y ago
    I'm asking about the actual properties of the equations, not if it's hard to do simulations.
davnn 4y ago
> This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. What kinds of problems does it solve to know an answer to this question? Honestly curious, please do not take this as offensive/dismissive.
- wwalker3 4y ago
  If mathematicians could solve these kinds of problems, they could answer valuable questions like "Will this equation always have a physically meaningful solution?" If the answer was "No", then we would know that the equation can't be a faithful model of reality. We already know that the incompressible Euler equations can't be a faithful model, for reasons I've mentioned elsewhere in the thread. But I think the hope is that if they can answer these questions for incompressible Euler, then they can eventually extend their techniques to more complex fluid equations like Navier-Stokes, which people generally assume (but can't yet prove) is physically reasonable. Simulation has great practical value, but it doesn't give you any guarantees about the behavior of the solutions for all the cases you haven't actually tried.
acchow 4y ago
The infinite implied speed is of course because the Euler equations are just a rough approximation of what is happening, right? Much like Newtonian physics was an approximation? There’s a real causality bound of the speed of light
- wwalker3 4y ago
  If you use the theory of nonlinear partial differential equations to analyze the behavior of compressible materials, you can find what are called the "characteristic" speeds, which are the speeds that various types of waves propagate at. Compressible materials tend two have two different characteristic speeds, one for sound waves and one for shock waves. The speed of sound basically works out to speed = sqrt(stiffness / density). So as a material gets stiffer, the speed of sound goes up. An infinitely stiff (i.e. incompressible) material by implication would have an infinite speed of sound, though this can't happen in any real material. Shock waves travel faster than sound, at a speed related to the pressure difference across the shock. The greater the pressure difference, the faster the shock travels. So if you had an infinite pressure difference, you could have an infinitely fast shock wave, but again this can't happen in the real world. However, sound and shock speeds only apply to pressure waves in a material. Other influences like gravity and electromagnetism travel at the speed of light. So for example, if you're doing fluid dynamics for plasma, then you'll have a third characteristic speed, the speed of light, because of the charged nature of the material attracting and repelling itself. There are also more exotic characteristics, like the speed of a propagating combustion front in a flammable material. But when you get to this level you're no longer just solving one simple set of differential equations.
voidwtf 4y ago
This sounds similar in some ways to the discussion that arose between AlphaPhoenix and Veritasium YouTube channels concerning the speed of electrons along a closed loop of wire. https://youtu.be/2Vrhk5OjBP8 https://youtu.be/2Vrhk5OjBP8 With a relevant illustration at 9:41 at where I think these topics intersect conceptually.

nuclearnice1 4y ago

What a wonderfully informative and educational comment. Thank you. Would you also be able to shed some light on what a singularity is? It was not intuitive to me that incompressiblity should lead to a singularity. The article dances around the term: > At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.” > Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.

isoprophlex 4y ago
I think (not a physicist), simply put, an infinity or NaN value. As these are step-wise methods, having such values show up anywhere will seriously mess up subsequent calculation steps.
prolyxis 4y ago
A simple example of a function with a singularity is f(t)=1/t. Note that at t=0, f(t) is undefined due to division by zero. On either side of zero, the absolute value of f(t) approaches infinity. In this case, we are tracking the flow of an incompressible fluid over time. This flow is represented by a velocity field evolving over time, under the constraint of no net inflow/outflow of material into any region of space. Thus, the singularity corresponds to a portion of fluid speeding up and approaching an infinite speed as you approach some finite time. Because the fluid cannot be compressed, the only way the singularity can be produced is for a portion of the liquid to swirl, increasingly rapidly, about some point: hence the discussion in the article about vorticity. As isoprophlex pointed out, this undefined value of the velocity field prevents you from (or at least complicates) computing the further evolution of the fluid.
- nico 4y ago
  Thank you for the great explanation. Do these swirls shed energy? Is it considered in these equations that for example friction within the swirls would slow them down (and hence not reach a singularity)?
  - pas 4y ago
    In real fluids yes, absolutely, they basically transform/branch/divide/split into smaller and smaller scale vortices and then those dissipate the energy into the fluid (heating it up a bit).
wwalker3 4y ago
The incompressible Euler equations model a fluid as a two-valued field. This means that at every point in space, the field has two values, density and velocity (1). To me (2), a singularity in a field like this means that one or more of the field values "blows up", i.e. goes to infinity as you run the time variable forward. But how could this ever happen? The Euler equations model the "conservation" (i.e. constant-ness) of three real physical quantities: mass, momentum, and energy. If these three quantities are finite and constant when you add them up over the whole field, how can any part of it "blow up" into an infinite value? The answer is that the blow-up must occupy a volume that shrinks as the blow-up grows, so the conserved quantities are still constant. The singularity would be infinitely small in space, and have an infinite value of density or velocity (or both). The hard question is, are these blow-ups merely artifacts of a particular numerical simulation technique, or are they essential somehow to the incompressible Euler equations themselves? That's what these papers are trying to figure out. To me, an "essential" (i.e. inherent-in-the-equations) blow-up seems intuitively reasonable because of the acausal nature of the field. When you simulate the incompressible Euler equations, it superficially looks like it's a physical fluid doing physical-fluid things, swirling and flowing around. But in a real fluid, a change in one part of the fluid propagates to the other parts at finite velocity, creating real cause and effect. An Euler fluid's time evolution is not a phenomenon that ripples forward through time in a normal way. Instead, every point in the fluid responds to every other point simultaneously. If you poke a cube of incompressible Euler fluid with your finger, there is no pressure wave that ripples through it, where the fluid parcels push each other along and get out of each other's way. Instead, the whole cube of fluid somehow instantly adopts a new flow pattern that conserves mass/momentum/energy in response to that finger-poke. 1) Note that velocity is a vector, since it has a direction. This means that in 2D the velocity is two numbers, and in 3D it's three numbers. So technically the 3D incompressible Euler equations have four values at every point: one density, and three velocity components, one each in the x, y, and z directions. 2) I'm a numerical simulation guy, not a mathematician. Real math experts have rigorous definitions of a singularity, e.g. in https://arxiv.org/pdf/2203.17221.pdf https://arxiv.org/pdf/2203.17221.pdf "Singularity formation in the incompressible Euler equation in finite and infinite time," Theodore D. Drivas and Tarek M. Elgindi.
- vba616 4y ago
NotYourLawyer 4y ago
It’s when some physical quantity of the simulation becomes infinite. Pressure, particle velocity, etc.
- TEP_Kim_Il_Sung 4y ago
  That usually indicates a phase change, with a separate set of applicable equations.

pencilguin 4y ago

I guess this answers why you can't just try it with liquid helium: even that isn't ideal enough.

WalterBright 4y ago

Best comment I ever read on HN.

wwalker3 4y ago
Wow, I'm honored :) These days, I try to only comment when an article is really in my wheelhouse, but that's not very often, given my narrow interests in fluid dynamics and computational physics.
- WalterBright 4y ago
  > I try to only comment when an article is really in my wheelhouse Which is why your comment is exceptionally worthwhile. I also know enough about fluid mechanics to both understand and appreciate it. People often wonder on HN what the point of a STEM degree is (after making money). To me I've had a lifetime of pleasure from understanding how things work. It's so much better than things being mysterious black boxes. I once asked a date if she wanted to understand how airplanes worked. She said no, that understanding them would make her afraid of flying. For me, it was the opposite. Knowing how the airplanes fly and how it all works made me a much less anxious passenger.

graycat 4y ago

Yes, in pure/applied math, we know a lot about various cases of approximation. But in practice there are more cases of approximation, and, right, the Euler equations are another such case. Or, to be a little flippant, generally in applications to real problems, we look at a lot of the features and throw out some, modify some, and actually honor some!! So, a question is, can we improve our ability to make such approximations and know something about the accuracy of the solutions we will get? E.g., for the Euler equations, will that approximation of an "incompressible" fluid ever work in practice and, if so, when and, there, how accurate can/will it be? Or, what about, hmm, just to be picky and pick something, friction on the side of the tube? What if the tube is not a perfect tube? A few grains of dirt: What if the liquid is water but, like most real water, has some solids floating around in it? Right, we can say, so there are a few grains of dirt floating around in the water, and they won't matter -- to be picky, that's an approximation, and we are likely correct, but where is an actual math theorem that says we are correct or how correct, i.e., accurate, are we? Right, a few grains of dirt -- we don't much care. But that's practical judgment and not really theorem/proof math. And similarly for other approximations we get as we throw out, modify, or honor real features? So, as stated, this is too difficult as a pure/applied math research direction. Okay, ..., then, is there anything at all in that direction that might be not absurdly difficult as a research direction? Or, to be simplistic, we work hard and get a numerical solution to a boundary value problem. Now someone tweaks the boundary. Can we say that our numerical solution is only tweaked? Or, when can we say that small changes in the problem statement will result in only small changes in the solution? Right, we are into some topology and looking for a case of continuity .... Hmm .... If we had some linearity ...!!! Right, the two pillars of analysis are continuity and linearity ...! But here with Euler we were considering nonlinear partial differential equations! Again I ask, is there any hope we can do anything for some corresponding math??

civilized 4y ago

This raises a question I hadn't thought of before. Real-world fluid flow is ultimately well-modeled by the equations of many-body Newtonian mechanics, right (atoms bumping around)? Are those equations vulnerable to blow-ups?

vba616 4y ago
This makes me think of: https://en.wikipedia.org/wiki/Sonoluminescence https://en.wikipedia.org/wiki/Sonoluminescence I would think that nothing in reality is infinite, but allegedly sound waves collapsing bubbles in a fluid can cause a very small amount of plasma to become hotter than the sun and emit light. Some controversial research claims it might even be possible to create atomic fusion this way.
wwalker3 4y ago
Pretty much any mathematical model of a real phenomenon can have some sort of singularity or discontinuity in it. If you model atoms as dimensionless points (1), then any kind of force law with the distance between atoms in the denominator can lead to a singularity when that distance is zero. In practice, you write the simulator to disallow this, but it's still there in the equations, you're just ignoring it. If you model your atoms as finite-sized but incompressible billiard balls, then when they hit each other it's a discontinuity, since they instantly change direction when they collide. These collisions conserve total momentum and energy, but they're unphysical because real physical quantities can't jump from one value to another (in classical physics). Even if you model your atoms as little rubber balls, the model can still be singular. Linear elasticity (the most common choice) allows you to compress a finite-sized object down to zero size with finite energy, which yields infinite energy density. Again, you'd have to disallow that in the simulator, which is very practical, but not theoretically satisfying. 1) https://en.wikipedia.org/wiki/Molecular_dynamics https://en.wikipedia.org/wiki/Molecular_dynamics is the typical method of atomistic simulation. 2) https://en.wikipedia.org/wiki/Linear_elasticity https://en.wikipedia.org/wiki/Linear_elasticity
- civilized 4y ago
  I'm asking about the actual properties of the equations, not if it's hard to do simulations.

davnn 4y ago

> This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine. What kinds of problems does it solve to know an answer to this question? Honestly curious, please do not take this as offensive/dismissive.

wwalker3 4y ago
If mathematicians could solve these kinds of problems, they could answer valuable questions like "Will this equation always have a physically meaningful solution?" If the answer was "No", then we would know that the equation can't be a faithful model of reality. We already know that the incompressible Euler equations can't be a faithful model, for reasons I've mentioned elsewhere in the thread. But I think the hope is that if they can answer these questions for incompressible Euler, then they can eventually extend their techniques to more complex fluid equations like Navier-Stokes, which people generally assume (but can't yet prove) is physically reasonable. Simulation has great practical value, but it doesn't give you any guarantees about the behavior of the solutions for all the cases you haven't actually tried.

acchow 4y ago

The infinite implied speed is of course because the Euler equations are just a rough approximation of what is happening, right? Much like Newtonian physics was an approximation? There’s a real causality bound of the speed of light

wwalker3 4y ago
If you use the theory of nonlinear partial differential equations to analyze the behavior of compressible materials, you can find what are called the "characteristic" speeds, which are the speeds that various types of waves propagate at. Compressible materials tend two have two different characteristic speeds, one for sound waves and one for shock waves. The speed of sound basically works out to speed = sqrt(stiffness / density). So as a material gets stiffer, the speed of sound goes up. An infinitely stiff (i.e. incompressible) material by implication would have an infinite speed of sound, though this can't happen in any real material. Shock waves travel faster than sound, at a speed related to the pressure difference across the shock. The greater the pressure difference, the faster the shock travels. So if you had an infinite pressure difference, you could have an infinitely fast shock wave, but again this can't happen in the real world. However, sound and shock speeds only apply to pressure waves in a material. Other influences like gravity and electromagnetism travel at the speed of light. So for example, if you're doing fluid dynamics for plasma, then you'll have a third characteristic speed, the speed of light, because of the charged nature of the material attracting and repelling itself. There are also more exotic characteristics, like the speed of a propagating combustion front in a flammable material. But when you get to this level you're no longer just solving one simple set of differential equations.

voidwtf 4y ago

This sounds similar in some ways to the discussion that arose between AlphaPhoenix and Veritasium YouTube channels concerning the speed of electrons along a closed loop of wire. https://youtu.be/2Vrhk5OjBP8 https://youtu.be/2Vrhk5OjBP8 With a relevant illustration at 9:41 at where I think these topics intersect conceptually.