Editing Partial differential equation (section)

== Theoretical Studies ==
As a branch of pure mathematics, the theoretical studies of PDEs focus on the criteria for a solution to exist, the properties of a solution, and finding its formula is often secondary.

=== Well-posedness ===
{{main|Well-posed problem}}
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
* an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
* by [[continuity (mathematics)|continuously]] changing the free choices, one continuously changes the corresponding solution
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.

=== Regularity ===
{{Main|Regularity theory}}
Regularity refers to the integrability and differentiability of weak solutions, which can often be represented by [[Sobolev space|Sobolev spaces]]. 

This problem arise due to the difficulty in searching for classical solutions. Researchers often tend to find weak solutions at first and then find out whether it is smooth enough to be qualified as a classical solution.

Results from [[functional analysis]] are often used in this field of study.