Why are weak solutions to PDEs good enough?

Solution 1:

Usually there are sufficiently nice regularity results. Sobolev spaces are often a useful way to show that a weak solution exists, where previously existence theory was harder to arrive at (although there are other techniques).

In other cases, the Sobolev space is sufficient. For instance, in the case of the heat equation, often it is desired to know how much heat energy is in the tank at a given time, what the flux of heat energy out of the domain is, how rapidly it will lose the energy it had to begin with, etc. These are all things for which a $L^2$ in time and $H^1$ in space solution are sufficient. In physical applications, rarely is the specific point value of a function important - it is averages over small patches in both time and space, which are integrals.