In the case of Navier-Stokes, this was done in 1934 by Jean Leray, who showed that the so-called 'weak' solutions always exist. (A weak solution is not a function, but a more general object called a distribution which can be fuzzed out on very short distance scales.)

It's a very reasonable idea, and useful enough for computations, but it lacks the essential tension that makes Navier-Stokes a mathematical Everest.