Comments on the question [Are those distributional solutions that are functions, the same as weak solutions?](https://mathoverflow.net/questions/366411/are-those-distributional-solutions-that-are-functions-the-same-as-weak-solution?noredirect=1#comment925399_366411) suggest there might not be a standard definition of the weak form of a non-linear PDE.


Is there one?


For my most specific concern: Is there a standard weak form for the Navier-Stokes equation (in the [Millennium Prize version](https://www.claymath.org/sites/default/files/navierstokes.pdf) if that matters)?



Is there a standard definition of weak form of a nonlinear PDE?
The study of nonlinear PDEs is almost always done in an ad hoc way. This is in sharp contrast to how research is done in almost every other area of modern mathematics. Although there are commonly used techniques, you usually have to customize them for each specific PDE.


