10.06.06

Penny Smith’s Proof on the Navier-Stokes Equations

Posted in Computational Fluid Dynamics at 9:45 pm by Brooks

This week’s big news in fluid flow simulation is Penny Smith’s existence proof for smooth solutions of the Navier-Stokes equations. (See also this in-depth explanation of the proof, and various other mentions of it here, here, and here.)

(Update: Regrettably, the article has been withdrawn due to a “serious flaw”. The piece described below is still interesting, however.)

The question that she’s solving is this: Given an infinite three-dimensional domain with arbitrary initial conditions, does a smooth solution to the Navier-Stokes equations (which are the equations describing the flow of water, air, or other fluids) exist on that domain for all future time? This problem was considered one of the most important open questions in mathematics by the Clay Institute, who are offering a one million dollar prize for a solution. It’s of some physical relevance directly, since if it’s not true then either there are cases where the Navier-Stokes equations don’t describe physical reality or else physical reality can behave in surprising ways. The greater significance, though, is that it and the methods that are used to prove it will be useful in further mathematical understanding of this particularly difficult problem.

Even though this is an existence proof and not a constructive one — which means that it proves that a solution exists, but does not provide an analytical method for finding one — there are some interesting construtive portions to the proof. Specifically, she shows that the solutions to the Navier-Stokes equations can be expressed as the limit of a set of hyperbolic equations. There is some interesting physical and computational relevance to how that works, which I thought would be interesting to more mathematically-inclined readers.

To start with, the Navier-Stokes equations are differential equations in four variables — pressure (p) and three directional components of velocity (ui, for i=1, 2, or 3). They look like this:

(Note that, in this notation, each term has an implicit sum over indices that appear twice in the term; thus, the second term in the first equation and the first term in the second equation are implicitly summed over k. As i appears only once in each term of the first equation, there isn’t an implicit summation over it, and thus that represents a set of three equations, one for each coordinate direction.)

There are some interesting features of this set of equations. First, there are four equations and four unknown variables, so it’s a closed set. However, if we start matching up variables with equations on a one-to-one basis, we find that the first three equations match up neatly with the three components of velocity, leaving the last for the pressure — even though it doesn’t actually have pressure in it (much less a time-derivative of pressure)!

What happens, then, is that at any point in time, the first three equations describe the time derivative of the velocity, as a function only of the local field at the current time (that is, they’re hyperbolic). This means that in these equations information travels across the domain at a finite rate, which is a very nice feature computationally; one can compute a solution at each grid point without needing to know the solution everywhere else. It’s also handy mathematically.

The fourth equation, though, messes all of this up. What it says is that the pressure needs to be chosen such that this velocity condition is satisfied everywhere. Physically speaking, if the velocities would be trying to compress a certain spot, the pressure there gets increased to counteract that, so it doesn’t compress. This isn’t a local constraint, because it’s the derivative of the pressure that affects the velocities, and so a change in pressure in one place causes immediate changes throughout the whole domain — essentially, the speed of information transfer is infinite. In a computational simulation, the vast bulk of the time is usually spent taking care of the pressure.

So, how does Penny Smith’s analysis approximate this by a set of hyperbolic equations? There are two parts to that. The first part is that she needs a set of first-order equations for her analysis, and this contains a second order term. Fixing that is straightforward, we simply introduce more variables, and a mathematical identity. The (nine) new variables are the components of a “viscous stress tensor”, defined as:

(That’s nine equations, one for each combination of i and k). (Also, for the fluid mechanicians: it’s not quite the viscous stress tensor, I know; there are terms left out that sum to zero.) Then, the first three equations can be rewritten as

This gives a set of thirteen first-order differential equations and thirteen unknowns.Now, for the hyperbolic part. I’ll start with the pressure equation, since that’s the one that has historically been a problem, but in this first-order formulation the stress-tensor equations also are not hyperbolic. To be a hyperbolic equation, this needs to have a time derivative of pressure in it. So, suppose we just add one, multiplied by an arbitrary constant (denoted ?):

If ? is zero, this is the same as the initial equation, but if ? is greater than zero, this and the other three initial equations form a hyperbolic set. Thus, if we take a set of ? values that converge to a limit of zero, we get a set of solutions of hyperbolic equations which converge to a solution of the initial non-hyperbolic equations.It turns out that this has physical significance — what happens is that, if the velocities are trying to compress a certain spot, the pressure there still goes up, but not immediately, and so things do get compressed but only a limited amount. This is exactly what happens in “compressible” flows, such as air flows at high speed, so it’s a phsyically meaningful problem. In effect, there’s a “relaxation time” — a finite amount of time that it takes for the pressure to respond to a given input; this relaxation time is proportional to ?, and so what we’ve done is taken an equation with a zero relaxation time and given it a finite nonzero one.

This suggests a rather simple computational method: solve the problem with a small but non-zero value of ?, and thereby obtain a solution that’s close to the desired one. This used to be a fairly common method, because hyperbolic problems are relatively easy to solve, and it’s called “artificial compressiblity”. It works reasonably well in some cases, although as ? gets closer to zero, the rates of information transfer around the domain get faster, and so the equations take more work to solve computationally. And, of course, if you use a larger ?, the rates of information transfer get slower, and so the solutions aren’t as close to the real solution. But it’s reasonably good for some problems.

Now, in Penny Smith’s case with everything in first-order equations, there are some additional equations that aren’t in hyperbolic form, so she does exactly the same thing to those equations, adding a time-derivative term to those as well:

In the computational situation, this isn’t necessary; the stress tensor can be calculatated from local data and doesn’t have problems of global dependencies, so there’s no need to add the nonzero relaxation time to these equations. However, it’s important for simplifying the mathematical analysis. (And it’s an interesting question whether it might be useful in computations as well; there are some advantages to working with things that are mathematically simpler.)So, that’s pretty much what’s going on with that part of the analysis. From there, the proof then goes off into two additional — and far more complicated — parts. First, there’s a proof that smooth solutions to these sets of first-order hyperbolic equations exist for all time, given a sufficiently smooth set of starting conditions. Then, there’s a reference to a proof from 25 years ago by John Heywood, which showed that smooth solutions to the Navier-Stokes equations exist at least for a finite time, and some analysis that shows that his solutions can be used to provide appropriate starting conditions for the first half of the proof. And, finally, there’s a small bit to tie everything together and show that the fact that the hyperbolic equations have smooth solutions means that their limit is also smooth. QED.

(Note: Equations in this article were typeset with MimeTeX.)

24 Comments »

  1. penny smith said,

    October 6, 2006 at 9:54 pm

    RIght on target, Brooke.
    best
    Penny

    This was indeed my motivation for the approximation.

  2. Brooks said,

    October 6, 2006 at 10:11 pm

    Thanks for the comment, Penny! It’s interesting seeing how ideas from one perspective on things turn out to be useful in a quite different one.

  3. Nonoscience / Analytical Solution to the Navier Stokes Equation Reported said,

    October 7, 2006 at 5:19 am

    [...] The above paper deals with an existence proof rather than a constructive one but nevertheless is a great effort. Brooke Moses has written an explanatory blog post on this and an analysis of greater rigour is also available. A nice discussion is also available with comments from Dr. Penny Smith at Peter Woit’s blog and a separate webpage in the Lehigh University maintained by Christina Sormani also is devoted for discussions. [...]

  4. A Winter’s Tale… » Existence of Smooth Solutions to Navier-Stokes said,

    October 7, 2006 at 2:06 pm

    [...] The big math news is that Penny Smith of Lehigh University appears to have proved the existence of smooth solutions to the Navier-Stokes equations. This is one of the ten Millenium Problems for which the Clay Mathematics Institute is offering a million-dollar prize. Brooks Moses has written a very illuminating non-technical explanation of Dr. Smith’s work. [...]

  5. Another Millenium problem solved?. - Page 2 - Math Help Forum said,

    October 7, 2006 at 4:20 pm

    [...] Also read about it here: Brooks Moses: Notes on Divergent Simulations » Penny Smith’s Proof on the Navier-Stokes Equations and [...]

  6. gs said,

    October 7, 2006 at 7:21 pm

    Brooks, please indulge a couple of questions although it’s been decades since I was involved in research.

    You write that “…a change in pressure in one place causes immediate changes throughout the whole domain — essentially, the speed of information transfer is infinite.” I thought that hydrodynamic information propagates at the speed of sound. What am I not understanding?

    I’d also like to follow up the uniqueness issue that Arunn mentions in his Nonoscience blog. Arunn writes, “…does this imply the particular solutions of NS equations may not be “physcial” or vice versa?” If Smith’s solution is not unique, it may not be stable; if it is not stable, might it not be unstable in favor of an as-yet-uncharacterized solution?

  7. Brooks said,

    October 7, 2006 at 7:40 pm

    gs: The first one’s easy. You’re right that hydrodynamic information propogates at the speed of sound. The only thing that you’re missing is that, in an incompressible fluid, the speed of sound is infinite. In the real world, there are of course no completely incompressible fluids, and so the speed of sound is always finite, but for water under most conditions (and even air, in a lot of cases), it’s reasonable to use an incompressible, infinite-speed-of-sound approximation to the equations. And that’s what the traditional Navier-Stokes equations are.

    I’ll have to think about the second question a bit to answer it completely. However, I’m reasonably certain that the solutions to classical initial-value problems such as this are always unique. (The multiple-solution instability thing is when there are multiple solutions to the steady-state problem, or sometimes to the time-periodic problem, which is a quite different thing.)

  8. Arun said,

    October 8, 2006 at 7:15 am

    What I think one would want to know, physically speaking, is whether over time a flow of a physical fluid that is initially well within in the incompressible regime require a change to the compressible regime? i.e., do the immortal solutions imply a upper bound on the pressure?

  9. Brooks said,

    October 8, 2006 at 7:20 pm

    Arun: In the general case, there’s certainly no guarantee of an upper bound — in the nondimensionalization, pressure gets scaled by the length- and time-scales used to do the nondimensionalizing, and so for a solution that gives a pressure increase P, one can easily produce a problem with the same nondimensionalized form but faster velocities and a different lengthscale, which will have a pressure increase of n*P for arbitrary n. (Or, at least, I think that’s true.) Thus, there’s no general guarantee.

    In the more interesting case, I guess at this point it’s an open question whether a proof of this sort will be able to impose a clear bound on the pressure as a function of the initial conditions. Penny’s didn’t, I believe.

  10. Arunn said,

    October 8, 2006 at 9:04 pm

    Brooks (and gs and Arun): Firstly, just to make it clear, if at all there was any slight doubt; I am Arunn, different from Arun who also seems to be discussing our pet problems and share the same field… ;)

    It is unfortunate that the paper is taken off. But I agree with Brooks that nevertheless what is discussed here is worthwhile. Also, isn’t this pseudo-comprressibility method the bread and butter of LBM?

  11. Brooks said,

    October 8, 2006 at 9:39 pm

    Arunn: Artificial compressiblity is, indeed, how all of the Lattice-Boltzmann methods that I have seen work — as of yet, nobody has developed a LBM that can deal with anything other than hyperbolic problems.

    There seem to be a number of variants on LBMs that have clever enhancements to the artificial-compressiblity equations to do a better job of modeling incompressible flows, but as far as I can tell from a quick look through the literature, they’re all still essentially solving hyperbolic equations with a finite rate of information propogation — there aren’t any methods that I’ve seen that do a complete many-iteration relaxation at each timestep (which is how you’d do a non-hyperbolic problem with a system like that).

    Smoothed-particle hydrodynamics models also use approximations like this, and with those I find it rather more surprising that I haven’t seen a relaxation-at-each-timestep incompressible method. But maybe there is such a thing and I just haven’t seen it.

  12. Arunn said,

    October 8, 2006 at 10:23 pm

    Brooks: Yes LBM uses always this artificial compressibility. I am just into it recently to simulate porous medium flows, my area of research interest wherein I encountered this. But I am yet to get a hands on on this; just yet fumbling with the LBM code…

  13. Sun said,

    October 9, 2006 at 8:47 am

    Brooks: Thanks for the nice explanation of the N-S equation. Just wonder how a unique solution of N-S equation describe a turbulent flow…

  14. Brooks said,

    October 9, 2006 at 9:31 am

    Sun: As with any chaotic system, unique solutions can be extremely sensitive to initial conditions, and still be unique for a given initial condition. Thus, given a particular set of initial conditions, there is only one set of turbulent eddies that will result — but, if that set of initial conditions is perturbed by a tiny amount, the perturbation will grow, and by the time turbulence develops, the eddies will be quite different.

    You won’t see this in experiments, of course, since it’s impossible to get exactly the same initial conditions twice. But it does show up in computations, unless you introduce randomness to avoid it.

  15. Sun said,

    October 9, 2006 at 10:58 am

    Brooks: You are right. In reality, there is disturbance all the time.

  16. Nonoscience / Philosophia Naturalis Part Deux said,

    October 12, 2006 at 9:00 am

    [...] The other one is more recent and is about the withdrawn paper of Dr. Penny Smith on the proposed existence solution for the Navier Stokes equations. Instead of directing us to the more sensational blog posts on this topic, we shall direct us to Penny Smith’s Proof on the Navier-Stokes Equations, a post by Brooks Moses at his Notes on Divergent Simulations blog, which exlpains the technical possibilities of such a scheme of transforming the NS equations into a hyperbolic equivalent and the implications. [...]

  17. gs said,

    October 15, 2006 at 6:07 am

    Brooks, thanks for your clarification on October 7. Good luck in your search for a postdoc.

  18. Alexandr Kozachok said,

    October 24, 2006 at 12:26 am

    Dear participants of discussion !
    You can detect unreasonableness in Penny Smith’s calculations if you will look the reference ” NAVIER -STOKES Equations Paradoxes ” on a site http://continuum-paradoxes.narod.ru . In detail look “Russian Pages”.
    Alexandr Kozachok

  19. Recovering my youth said,

    April 22, 2007 at 2:09 pm

    [...] UPDATE: I finally found a really nice explanation of the problem on this blog which attempts to explain a proposed (but now shown wrong) solution. Read up to but not including the paragraph beginning “So, how does Penny Smith’s analysis approximate this by a set of hyperbolic equations?” [...]

  20. Dmitri Gorskin said,

    December 4, 2007 at 2:53 am

    I found exact analytical solution for 3 D viscous nonsteady flow(without cappilar effect) many years ago.But I never thought,that it is nesessary to prove existence this smooth solution for small period of time.It is clear, like a day.
    But when I start research it, I was surprised.The water has two properties, like living object(like cell).Competition and reproductions some of his parameters.It was amazing.

  21. Alexandr Kozachok said,

    December 15, 2007 at 1:22 am

    Dear Dmitri Gorskin!
    Where your “exact analytical solution” is published?
    Alexandr Kozachok

  22. Alexandr Kozachok said,

    February 16, 2008 at 7:36 pm

    MILLENNIUM PRIZE PROBLEM (NAVIER– STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS

    Kozachok A.A., Kiev, Ukraine

    Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier – Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem’s solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description–Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution’s obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
    It is known, divu and divv are identical to infinitesimal magnitude and velocity of the relative modification of a volume element of the strained medium. Therefore divergency of acceleration divw, probably, there is a magnitude, identical to a acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements divu=0, divv=0 it is necessary to accept divw=0.
    The requirement divw=0 for incompressible liquid is formulated by analogy and proved. Operation div will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure p=p(x,y,z,t) at some limitations of a mass force vector. The time enters into Laplace equation as parameter.
    Laplacian of the Navier – Stokes equations (pressure p=p(x,y,z,t) – harmonic function) and a change of a variable (velocity on acceleration) allow to gain system of conventionally independent integro-differential equations with acceleration’s components w. In that case components of the acceleration for ideal incompressible liquid are harmonic functions too. The change of a variable allows to use boundary conditions of an adhesion of a fluid absolutely correctly. According to this requirement vectors of acceleration on firm immobile boundary line are equal to null.
    Conversion of the Navier-Stokes equations to more simple equations has actually removed a problem of an existence proof and smoothness of their solution. It is possible to use known effects about properties of harmonic functions or representation of the common decision of the Laplace equation (http://continuum-paradoxes.narod.ru, the link « Manual, p.1 », p. 58). More in detail on a site http://continuum-paradoxes.narod.ru, the link “Russian pages”, “Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods (in Russian)”.
    Yours faithfully, Alexandr Kozachok

  23. a-kozachok1 said,

    March 15, 2008 at 2:59 am

    Dear Brooks !
    I very hope, that you and your colleagues from US universities will formulate the comment in this occasion:
    MILLENNIUM PRIZE PROBLEM (NAVIER– STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS.
    The Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
    1. It is known, divu and divv are identical to infinitesimal magnitude and velocity of the relative modification of the infinitesimal volume. Therefore divergency of acceleration divw - acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements divu=0, divv=0 is necessary to accept divw=0.
    2.The requirement divw=0 for incompressible liquid is formulated by analogy and proved. Operation div will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure p=p(x,y,z,t) at some limitations of a mass force vector. The time t enters into Laplace equation as parameter.
    Read more on http://a-kozachok1.narod.ru ,“Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods” (Russian).

    Yours faithfully, Alexandr Kozachok

  24. Nonoscience / Philosophia Naturalis Part Deux said,

    July 28, 2008 at 9:17 am

    [...] Instead of directing us to the more sensational blog posts on this topic, we shall direct us to Penny Smith�s Proof on the Navier-Stokes Equations, a post by Brooks Moses at his Notes on Divergent Simulations blog, which explains the technical [...]

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