TITLE: Have people successfully worked with the full ring of differential operators in characteristic p?
QUESTION [26 upvotes]: 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).

REPLY [4 votes]: 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)