Navier-Stokes is a non-linear PDE, and there is no standard, general theory of weak solutions for nonlinear PDEs. But the literature on weak solutions to the incompressible Navier-Stokes constantly uses a specific weakening of the equation which has this nice property: any smooth function which is a weak solution (in the sense of this weak form) is in fact a strong solution. This is much less informative than weak-strong uniqueness because it says nothing about uniqueness. But in return, it is trivial to prove.
My question is: Is there a standard term for describing those weak forms of PDEs such that every weak solution which is also a smooth function is a strong solution?
In case that is unclear I will give more detail. The unforced Navier Stokes equation is often expressed as $$\partial_t \mathbf{u} + \mathbf{ u} \cdot \nabla \mathbf{u} = \nu\cdot\nabla^2 \mathbf{u} -\nabla p .$$ But when $\mathbf{u}$ is smooth, and we assume incompressibility, this is equivalent to $$\partial_t \mathbf{u} + \nabla (\mathbf{ u} \otimes \mathbf{u}) = \nu\cdot\nabla^2 \mathbf{u} -\nabla p .$$
This latter equation is still nonlinear but it can be weakened by the same, standard procedure used to weaken linear PDEs. Integrate it against any test function, using formal integration by parts to remove the derivatives from $\mathbf{u}$ and $\nabla (\mathbf{ u} \otimes \mathbf{u})$ and put them on the test function. So any divergence free smooth function that solves all those integral equations, solves the original PDE.
