The Navier-Stokes system for incompressible fluids in $\mathbb R^3$
reads as
\begin{align}
&\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=0,
\\
&v(t=0)=a,\quad \text{div} a=0,
\end{align}
where $\mathbb P$ is the Leray projector.
Let us assume that $v$ and $w$ are two $C^\infty$ solutions of the above system, having in particular the same initial datum $a$.  

Question 1: Is it absolutely clear that $v=w$, if $v,w$ both belong to $L^r_{t,x}$ for some finite $r$? A classical result asserts that uniqueness holds true provided one of the solutions belongs to $L^p_tL^q_x$ with 
$
\frac2{p}+\frac3{q}=1,
$
for instance $L^2_tL^\infty_x$. But assuming smoothness for both $v$ and $w$ does not 
encompass a global estimate of that kind, so this classical result seems to fall short of providing a proof of the previous uniqueness result.

Question 2: A grand open problem is to ask for the uniqueness of Leray solutions, i.e. belonging to 
$L^\infty_tL^2_x\cap L^2_t\dot H^1_x$ with an initial datum in $L^2$: if $v,w$ are two Leray solutions which are also $C^\infty$ it does not seem clear that you can prove that  $v=w$ although you have made the drastic smoothness assumption and in doing so, ruled out turbulent solutions.