TITLE: Why should I "believe in" weak solutions to PDEs?
QUESTION [100 upvotes]: This is a sort of soft-question to which I can't find any satisfactory answer. At heart, I feel I have some need for a robust and well-motivated formalism in mathematics, and my work in geometry requires me to learn some analysis, and so I am confronted with the task of understanding weak solutions to PDEs. I have no problems understanding the formal definitions, and I don't need any clarification as to how they work or why they produce generalized solutions. What I don't understand is why I should "believe" in these guys, other than that they are a convenience.
Another way of trying to attack the issue I feel is that I don't see any reason to invent weak solutions, other than a a sort of (and I'm dreadfully sorry if this is offensive to any analysts) mathematical laziness. So what if classical solutions don't exist? My tongue-in-cheek instinct is just to say that that is the price one has to pay for working with bad objects! In other words, I do not find the justification of, "well, it makes it possible to find solutions" a very convincing one. 
A justification I might accept, is if there was a good mathematical reason for us to a priori expect there to be solutions, and for some reason, they could not be found in classical function spaces like $C^k(\Omega)$, and so we had to look at various enlargements in order to find solutions. If this is the case, what is the heuristic argument that tells me whether or not I should expect a PDE (subject to whatever conditions you want in order to make your argument clear) to have solutions, and what function space(s) are appropriate to look at to actually find these solutions?
Another justification that I would accept is if there was some good analytic reason to discard the classical notion of differentiability all together. Perhaps the correct thing to do is to just think of weak derivatives as simply the 'correct' notion of differentiability in the first place. My instinct is to say that maybe weak solutions are a sort of 'almost-everywhere' type generalization of differentiability, similar to the Lebesgue integral being a replacement for the Riemann integral which is more adept at dealing with phenomena only occurring in sets of measure $0$.
Or maybe both of these hunches are just completely wrong. I am basically brand new to these ideas, and wrestling with my skepticism about these ideas. So can somebody make me a believer?
Worth noting is that there is already a question on this site here, but the answer in this link is essentially that there exist a bunch of nice theorems if you do this, or that physically we don't care very much about what happens pointwise, only in terms of integrals over small regions. It should be clear why I don't like the first reason, and the second reason I may accept if it could be turned into something that looks like my proposed justification #2 - if integrals over small regions of derivatives are the 'right' mathematical formalism for PDEs. I just don't understand how to make that leap. In other words, I would like a reason to find weak solutions interesting for their own sake.

REPLY [6 votes]: The existing answers provide good reasons towards the question in the title, but from the perspective of a geometer I feel the applications in physics aren't quite as convincing. It's true that singular phenomena that arises in for example conservation laws requires a suitable notion of a generalised solution, but why is it also useful for geometric problems?
One way I think of weak solutions is that they provide a candidate for a strong solution. Suppose you want to a solve a particular PDE problem with suitable data and you can prove the following:

A weak solution exists.
Any classical solution, if it exists, is also a weak solutions.
The weak solution is suitably unique.

Then from the above you can infer that if a classical solution exists, it must be the unique weak solution. Hence the problem of existence is effectively reduced to proving the regularity of the weak solution.
Hence in nice cases where existence can established in general (e.g. linear elliptic problems), weak solutions provide a way of solving PDE problems using the above methodology. This is method is effective for the technical reason that it allows us to work in spaces with better compactness properties.
If a solution doesn't always exist however, things get more interesting. If you can still establish the first three points, the solubility criterion is reduced to a regularity problem and we can then look for necessary/sufficient conditions based on this.
Example (Harmonic map flow): If $(M,g)$ and $(N,h)$ are Riemannian manifolds, a classical problem in geometric analysis is whether a non-trivial harmonic map $u : M \rightarrow N$ exists. In the case when $M$ is a closed surface, we have the following sufficient condition for existence due to Eells and Sampson; non-trivial harmonic maps $M \rightarrow N$ exist provided there exists no non-trivial harmonic map $S^2 \rightarrow N.$
This theorem can be proved using the harmonic map flow to "evolve" a given map $u_0$ into a harmonic map $u_*,$ which is the work of Struwe. This method doesn't always work as the flow may develop singularities in general, but the non-existence condition about harmonic spheres provides a sufficient condition to prevent these singularities from forming.