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 $a$. Additionally we suppose that $v,w$ both belong to $L^r_{t,x}$ for some finite $r$.

Question: Is it absolutely clear that $v=w$? 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.