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. this survey 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 $0$.