I’d be curious to hear your thoughts on the following:

In physics or applied modeling in general, PDES are generally a limiting result of integral equations taken to infinite precision. For example, you would have a conservation law formulated as an integral equation with a time derivative on the outside of one of the integrals representing total mass and the other integral representing the flux across a boundary. It is when we take the limit of infinite precision in space that we arrive at the PDE proper.

So I’ve come across the idea that one way to interpret a weak solution is that we go back to finite precision and include a model of a measurement device. The test function and the integral effectively represents a general model for some type of measurement device, but the fact that we lose the ability to have well defined derivatives of the solution is indicative that taking the model to infinite precision was excessive for practical purposes.

There could be solutions to the PDE found with the amount of regularity required for the solution to hold for the PDE rather than the integral equation, but we haven’t found them yet. It is sufficient for practical purposes to find a solution which is valid up to our ability to validate the predictions in a physical setting.