Have people successfully worked with the full ring of differential operators in characteristic p? This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$.  (Here by the full ring of differential operators I mean the same thing as the ring of divided-power differential operators, which is the terminology used in the cited question.)
My question is:

Do people have experience using the full ring of differential operators successfully in characteristic $p$ (for localization, or other purposes)?

I always found this ring somewhat unpleasant (its sections over affines are not Noetherian,
and, if I recall correctly a computation I made a long time ago, the structure sheaf
${\mathcal O}_X$ is not perfect over ${\mathcal D}_X$).  Are there ways to get around
these technical defects? (Or am I wrong in thinking of them as technical defects, or am
I even just wrong about them full stop?)
EDIT: Let me add a little more motivation for my question, inspired in part by Hailong's answer and associated comments. A general feature of local cohomology in char. p is that you don't have the subtle theory of Bernstein polynomials that you have in char. 0. See e.g. the paper of Alvarez-Montaner, Blickle, and Lyubeznik cited by Hailong in his answer. What I don't understand is whether this means that (for example) localization with the full ring of differential ops is hopeless (because the answers would be too simple), or a wonderful prospect (because the answers would be so simple).
 A: Smith and Van den Bergh worked with it, and got some lovely results, in this paper.1  For example, they show that direct summands of polynomial rings have simple rings of differential operators in positive characteristic.   This is still open (as far as I know) in characteristic 0.
It's a particularly interesting paper for the connections it makes with representation types.
1Smith, Karen E.; Van den Bergh, Michel, Simplicity of rings of differential operators in prime characteristic, Proc. Lond. Math. Soc., III. Ser. 75, No. 1, 32-62 (1997). ZBL0948.16019, MR1444312.
A: This might not be what you are looking for, as they use the actual full ring of differential operators (in Berthelot's theory, your "full ring" would be $D^{(0)}$, if I understand correctly), but the following papers are very beautiful in my opinion:
Gieseker, D. - Flat vector bundles and the fundamental group in non-zero characteristics.
dos Santos, João Pedro Pinto - Fundamental group schemes for stratified sheaves. J. Algebra 317 (2007), no. 2, 691--713.
Hélène Esnault, Vikram Mehta - Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles
You'll of course notice very quickly that in all cases, the D-Module flavor is lost, as a $O_X$-coherent D-module can be translated into the world of vector bundles thanks to Frobenius descent.
A: Dear Matt: The people who are actively working with this whom I know are Genady Lyubeznik and Manuel Blickle.
The key point seems to be that certain $R[F]$-modules become simpler when viewed as $D_R$-modules (here $F$ is the Frobenius). It has been applied to show that certain local cohomology modules over regular local rings in positive characteristic have finitely many associated primes. Examples are in the following papers:

*

*D-module structure of R[F]-modules, Trans. Am. Math. Soc 355 (2003), 4, 1647–1668, and


*Generators of D-modules in positive characteristic, Math. Res. Lett. 12 (2005), no. 4, 459–473,
and the references therein. There is also this new preprint, Lyubeznik, Zhang and Zhang, A Property of the Frobenius Map of a Polynomial Ring, which might be of interest.
A: Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:
MR1918185 (2003h:14030)
Bögvad, Rikard(S-STOC)
An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary)
The Roos Festschrift volume, 1.
Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116.
14F10 (16S32 32C38)
