As I understand it, [Lions and DiPerna demonstrated existence and uniqueness](http://www.jstor.org/pss/1971423) for the [Boltzmann equation](http://en.wikipedia.org/wiki/Boltzmann_equation). Moreover, [this paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VJ2-44D2CY3-J&_user=10&_coverDate=11%252F30%252F2001&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1254807045&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=e923283d5d59dec6efcfaab5d995390c) 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?
> 
> 
> 



Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes?
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. 


