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 differenial 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).

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).

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 differenial 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).

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 differenial 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).

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 differenial 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?)

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 differenial 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).