Lawrence Evans wrote in discussing the work of Lions fils that
there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probabilistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods.
To me the second part of Evans' quote does not necessarily imply the first. So my question is: why can't there be a core theory of nonlinear PDE?
More specifically it is not clear to me is why there cannot be a mechanical procedure (I am reminded here by [very] loose analogy of the Risch algorithm) for producing estimates or good numerical schemes or algorithmically determining existence and uniqueness results for "most" PDE. (Perhaps the h-principle has something to say about a general theory of nonlinear PDE, but I don't understand it.)
I realize this question is more vague than typically considered appropriate for MO, so I have made it CW in the hope that it will be speedily improved. Given the paucity of PDE questions on MO I would like to think that this can be forgiven in the meantime.