I want to know if there is a good website which allows you to download survey papers on PDEs? The "survey" should include a summary of methods, skills, developments etc. I wish to get some basic (or common) conceptual ideas and methodology in the macroscopic view, as well as details. I think it can help me to understand the theory of PDEs and let me know how to do research in the field of PDEs? Any recommendation would be appreciated! ^_^
This aspects this question addresses is way too broad to be articulated in short, my suggestion, as a PhD student, overall would be: Take a graduate level course on PDE, instead of reading a survey paper.
Some lecture notes would equipped you with techniques developed in the recent two centuries, and methods of analyzing specific questions should give you the philosophy behind developing the delicacies for establishing some astounding results. For example, we learn all those beautiful results in Sobolev spaces, we may ask why, at first hand, to devise such mechanisms to solve the PDEs?
On the other hand, modern PDE theory evolves naturally from equations in physics, benefits from functional analysis(or you could say functional analysis originates from the idea of designing tools to analyze a class of PDEs), and shows beautiful differential geometric structures.
The study of PDEs spawns many new branches of applied mathematics, and I doubt a single survey paper would cover them all nicely in a higher philosophical point of view(if there exists such literature, I would like to get my hands on too), and even a great graduate level PDE textbook, like L.C. Evans, would merely tour you through a small fraction of this vast research area.
If you could share with us a specific equation you are interested in, I believe MSE would help you with more detailed directions.
An excellent survey article is "Partial Differential Equations in the 20th Century" by Haïm Brezis. The first part doesn't require much background, but you can enjoy it much more if you have already seen some basic PDE theory. A superbly written PDE book I can recommend is Folland's Introduction to Partial Differential Equations.
I second the recommendation of Evans' PDE textbook. Definitely the best PDE book I've read. PDEs of Applied Mathematics by Zauderer is also fairly good and comprehensive.