Uniqueness of Leray-Hopf weak solutions without forcing
In plain words
Weak solutions of the Navier-Stokes equations, which satisfy the equations only in an averaged sense, always exist, but it is unknown whether they are unique when no outside force acts. With a specially designed force, two different solutions from the same start are known.
Precise statement
3D Navier-Stokes on $R^{3}$ with $f = 0$ and divergence-free $u_{0}$ in $L^{2}$: is the Leray-Hopf weak solution (u in L^inf_t L^2_x and L^2_t H^1_x, satisfying the energy inequality) unique? Albritton, Brue and Colombo proved non-uniqueness for a force $f\ \text{in}\ L^{1}_{t} L^{2}_{x}$ built from an unstable self-similar profile. Answer: yes or no with proof.
What would settle it
A proof of uniqueness for all $L^{2}$ data, or two distinct Leray-Hopf solutions from the same data with $f = 0$.
Status in the literature
Non-uniqueness with forcing was shown by Albritton, Brue and Colombo (Annals of Mathematics, 2022); the unforced case remains open.