I was just wondering, reading Navier-Stokes official problem description (https://www.claymath.org/sites/default/files/navierstokes.pdf), if only one solution on $R^3$, respecting all required conditions, would be enough to answer the asked question in the official problem description. Because it is said that has be to proved the existence of, solutions… But do all solutions have to be found and described to fit with what is expected?
I didn’t find clear informations about that.
Sincerely.
For Navier-Stokes, it is known since the 60's that if a strong (i.e. smooth) solution exists, it is unique, even in the class of weak solutions with the same initial data. This property is called weak-strong uniqueness (see e.g. Wiedemann - Weak–strong uniqueness in fluid dynamics for more details).
What this means is that, once you found one smooth solution for given initial data, you will have found all solutions for that initial data.
Keep in mind however, that the official problem statement still requires you to show the existence of solutions for all possible choices of smooth initial data $u^0$. Otherwise the problem would be trivial, as $u\equiv 0$ clearly is a smooth solution of the Navier-Stokes equation with initial data $0$.
