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As I understand it, Lions and DiPerna demonstrated existence and uniqueness for the Boltzmann equation. Moreover, this paper claims that

Appropriately scaled families of DiPerna–Lions renormalized solutions of the Boltzmann equation are shown to have fluctuations whose limit points (in the weak $L^1$ topology) are governed by a Leray solution of the limiting Navier–Stokes equations.

Probably there is a lot of other work along these lines. But I am not well-versed enough in these areas to go through the literature easily, and so I hope someone can give a very high-level answer to my question:

Why does renormalizing the Boltzmann equation not (yet?) give existence and uniqueness for Navier-Stokes?

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  • $\begingroup$ Isn't everything you want to know already included in what you quoted? the rescaling limits in weak L^1 to Leray. Existence of Leray solutions to Navier-Stokes is already well-known, the problem is that we don't have regularity or uniqueness. If you are limiting in weak L^1, you lose regularity. If you are only talking about limit points, you can lose uniqueness. $\endgroup$
    – Willie Wong
    Mar 18, 2010 at 10:04
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    $\begingroup$ Lastly, I am not sure Lions and DiPerna proved what you think they did. I don't think they have a uniqueness result stated in there. Also they, like those around the time, assumed an angular cutoff property. Only recently (as in the past year or so) have results appeared that removes that assumption (while restricting to less general classes of interaction kernels). (See, e.g. 0912.0888, 0912.1426 on arXiv.) But this is probably only tangential to your question. $\endgroup$
    – Willie Wong
    Mar 18, 2010 at 10:25
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    $\begingroup$ To replace an earlier comment of mine: rather then linking to ScienceDirect, it is better to just give us the DOI number. The third link, as it currently is, is tied to your (or your institute's) ScienceDirect subscription, and when I click on it says something about invalid username. I assume you mean the paper of Golse and Saint-Raymond in the third link? springerlink.com/content/9d6yk5556q55fymc $\endgroup$
    – Willie Wong
    Mar 18, 2010 at 10:29
  • $\begingroup$ @Willie, Yes, I did mean that paper. I will leave the question unedited since your comment addresses it. Thanks for the DOI tip and also for your other comments and answer, which was really good. $\endgroup$
    – Steve Huntsman
    Mar 18, 2010 at 11:51
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    $\begingroup$ @TheAmplitwist :-( My bad. $\endgroup$
    – Willie Wong
    May 18, 2023 at 13:14

2 Answers 2

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Okay, after figuring out which paper you were trying to link to in the third link, I decided that it is better to just give an answer rather then a bunch of comments. So... there are several issues at large in your question. I hope I can address at least some of them.

The "big picture" problem you are implicitly getting at is the Hilbert problem of hydrodynamical limit of the Boltzmann equations: that intuitively the ensemble behaviour at the large, as model by a fluid as a vector field on a continuum, should be derivable from the individual behaviour of particles, as described by kinetic theory. Very loosely tied to this is the problem of global existence and regularity of Navier-Stokes.

If your goal is to solve the Navier-Stokes problem using the hydrodynamic limit, then you need to show that (a) there are globally unique classical solutions to the the Boltzmann equations and (b) that they converge in a suitably regular norm, in some rescaling limit, to a solution of Navier-Stokes. Neither step is anywhere close to being done.

As far as I know, there are no large data, globally unique, classical solutions to the Boltzmann equation. Period. If we drop some of the conditions, then yes: for small data (perturbation of Maxwellian), the recent work of Gressman and Strain (0912.0888) and Ukai et al (0912.1426) solve the problem for long-range interactions (so not all collision kernels are available). If you drop the criterion of global, there are quite a bit of old literature on local solutions, and if you drop the criterion of unique and classical, you have the DiPerna-Lions solutions (which also imposes an angular-cutoff condition that is not completely physical).

The work of Golse and Saint-Raymond that you linked to establishes the following: that the weak solution of DiPerna-Lions weakly converges to the well-known weak solutions of Leray for the Navier-Stokes problem. While this, in some sense, solve the problem of Hilbert, it is rather hopeless for a scheme trying to show global properties of Navier-Stokes: the class of Leray solutions are non-unique.

As I see it, to go down this route, you'd need to (i) prove an analogue of DiPerna-Lions, or to get around it completely differently, and arrive at global classical and unique solutions for Boltzmann. This is a difficult problem, but I was told that a lot of very good people are working on it. Then you'd need (ii) also to prove an analogue of Golse-Saint-Raymond in a stronger topology, or you can use Golse-Saint-Raymond to first obtain a weak-limit that is a Leray solution, and then show somehow that regularity is preserved under this limiting process. This second step is also rather formidable.

I hope this somewhat answers your question.

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  • $\begingroup$ Wow, thanks! Thanks also for educating me about non-uniqueness of the current methods for Boltzmann, which I was obviously ignorant of. $\endgroup$
    – Steve Huntsman
    Mar 18, 2010 at 11:42
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    $\begingroup$ Oh, lest I make a false impression: I didn't mean to say that the DiPerna-Lions solutions are non-unique (I am personally not aware of a counterexample, maybe there are some in the literature?); I just meant that their method does not prove uniqueness. $\endgroup$
    – Willie Wong
    Mar 18, 2010 at 11:58
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    $\begingroup$ Boltzmann equation explicitly ignores many-body collisions. So that's not the source. The difficulty is just an issue of the mathematics. More precisely, one of topology. Generally, one way to prove existence of solutions to PDEs is to construct a bounded sequence of functions in some high regularity function space, such that they approximately solve the equation you are looking at, with errors going down to zero in some low regularity norm. Then if the high regularity function space has a compact embedding into a long regularity space, you get for free a low regularity weak solution. $\endgroup$
    – Willie Wong
    Mar 18, 2010 at 16:36
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    $\begingroup$ But compactness arguments only guarantee "convergent subsequences", and not that the sequence converges as a whole. So in particular, they can be many limit points of the original bounded sequence in the weaker topology of the low regularity space. In this case, uniqueness will follow if you can show certain a priori control for the equation satisfied by the difference of two approximate solutions. This is hard for non-linear equations. [Often when such control can be had, one can start over using this control to run a contraction mapping argument from the get-go, getting existence AND uniq.] $\endgroup$
    – Willie Wong
    Mar 18, 2010 at 16:41
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    $\begingroup$ This entry serves as a great survey for current research in Boltzmann equations. Many thanks to Willie. $\endgroup$
    – John Jiang
    Jun 7, 2010 at 17:39
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There's been some recent work on the Boltzmann equation that's gotten a lot of press: see the first few links at http://www.math.upenn.edu/~strain/ (doi: 10.1073/pnas.1001185107 , http://www.math.upenn.edu/~strain/preprints/gsNonCut2.pdf ). I'm not sure if it's of interest to this discussion. I didn't even realize these issues hadn't been settled in the 19th century.

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    $\begingroup$ That's work in the same series as the one I mentioned in the fourth paragraph of my answer. :) $\endgroup$
    – Willie Wong
    May 24, 2010 at 12:44
  • $\begingroup$ @Willie: could you please say a few words about the progress made so far assuming the hard-sphere interaction potential? I am a bit confused about which one is harder, the hard-sphere or the long range interaction. $\endgroup$
    – John Jiang
    Jun 7, 2010 at 17:49
  • $\begingroup$ @John: I don't actually (currently) work on Boltzmann, so I am not the right person to ask about this. Most of what I know I pick up from conversations with experts who do. If you are really interested in the details I suggest you contact Bob Strain (UPenn), Yan Guo (Brown), Cedric Villani (IHP), or Clement Mouhot (CNRS/Cambridge). Unfortunately, as far as I know, none of them are on MO. $\endgroup$
    – Willie Wong
    Jun 8, 2010 at 13:57

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