most recent 30 from http://mathoverflow.net 2013-05-22T22:00:30Z http://mathoverflow.net/feeds/question/18539 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18539/why-dont-existence-and-uniqueness-for-the-boltzmann-equation-imply-the-same-for Steve Huntsman 2010-03-18T02:09:50Z 2010-05-24T03:53:55Z

<p>As I understand it, <a href="http://www.jstor.org/pss/1971423" rel="nofollow">Lions and DiPerna demonstrated existence and uniqueness</a> for the <a href="http://en.wikipedia.org/wiki/Boltzmann_equation" rel="nofollow">Boltzmann equation</a>. Moreover, <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6VJ2-44D2CY3-J&amp;_user=10&amp;_coverDate=11%252F30%252F2001&amp;_rdoc=1&amp;_fmt=high&amp;_orig=search&amp;_sort=d&amp;_docanchor=&amp;view=c&amp;_searchStrId=1254807045&amp;_rerunOrigin=google&amp;_acct=C000050221&amp;_version=1&amp;_urlVersion=0&amp;_userid=10&amp;md5=e923283d5d59dec6efcfaab5d995390c" rel="nofollow">this paper</a> claims that</p> <blockquote> <p>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.</p> </blockquote> <p>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: </p> <blockquote> <p>Why does renormalizing the Boltzmann equation not (yet?) give existence and uniqueness for Navier-Stokes?</p> </blockquote>

http://mathoverflow.net/questions/18539/why-dont-existence-and-uniqueness-for-the-boltzmann-equation-imply-the-same-for/18567#18567 Willie Wong 2010-03-18T10:57:37Z 2010-03-18T10:57:37Z

<p>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. </p> <p>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. </p> <p>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. </p> <p>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). </p> <p>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. </p> <p>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. </p> <p>I hope this somewhat answers your question. </p>

http://mathoverflow.net/questions/18539/why-dont-existence-and-uniqueness-for-the-boltzmann-equation-imply-the-same-for/25744#25744 none 2010-05-24T03:53:55Z 2010-05-24T03:53:55Z

<p>There's been some recent work on the Boltzmann equation that's gotten a lot of press: see the first few links at <a href="http://www.math.upenn.edu/~strain/" rel="nofollow">http://www.math.upenn.edu/~strain/</a> (doi: 10.1073/pnas.1001185107 , <a href="http://www.math.upenn.edu/~strain/preprints/gsNonCut2.pdf" rel="nofollow">http://www.math.upenn.edu/~strain/preprints/gsNonCut2.pdf</a> ). 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.</p>