I have noticed that two omissions are frequent in PDE theory, to the point that it has even cropped up in textbooks and very much discouraged my study of the field. This is my 5th attempt at learning the subject seriously, and I would like some advice, say in the form of answers to the following specific questions, thanks:
- Why is the solution space rarely specified? Even in ODE theory, we need to say "We are looking for solutions in the space of continuous functions." (Or more usually $C^k$ where $k$ is the order of the DEQ, although vacuously no function with not enough smoothness could solve an ODE in the traditional sense.)
- Speaking of the traditional sense as opposed to distributional or whatever else exists in PDE theory (I've heard terms like "weak solution"), how come none of this stuff showed up in ODE theory? In $d=1$ are all distributions functions? I am aware, without proof, of a result that says that all linear systems have a distributional solution if and only if it is, in some appropriate sense, a function, for $d=1$. What about other systems? Couldn't richer collections of solutions be obtained if we allowed distributions?
- In both PDE and ODE theory, I have noticed only half the problem is usually solved, in the sense that the authors usually take a given equation, and perform operations on it and end up at some "solution." But this solution need not actually work, especially if noninvertible operations are used, such as the Fourier transform as it's defined on $L^1$. In ODE theory, without allowing for the generalities in 2., it has usually been easy to verify a guessed solution indeed works. Although it may not be as trivial, is this what authors are assuming in 3.? Or is there some agreement among those in PDE theory that not all guessed solutions may actually work?