Uniqueness of Leray-Hopf weak solutions without forcing · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

Title
Uniqueness of Leray-Hopf weak solutions without forcing
Plain
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 · changed · core
Revision 1: 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.
Revision 2: 3D Navier-Stokes on $R^{3}$ with $f = 0$ and divergence-free $u_{0}$ in $L^{2}$: is the Leray-Hopf weak solution ($u\ \text{in}\ L^{\infty}_{t} L^{2}_{x}$ 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.
Settled by
A proof of uniqueness for all $L^{2}$ data, or two distinct Leray-Hopf solutions from the same data with $f = 0$.
Kind
well-posed
Answer type
yes-no
References
(empty)
Literature status
open
Status note
Non-uniqueness with forcing was shown by Albritton, Brue and Colombo (Annals of Mathematics, 2022); the unforced case remains open.
Posed since
1934

Decisions