Non-uniqueness conjecture for axially symmetric swirl-free Navier–Stokes solutions

Let $u$ solve the incompressible Navier–Stokes equations on $[0,1]\times\mathbb{R}^3$,

$$\partial_tu+u\cdot\nabla u+\nabla p-\Delta u=0,\qquad \nabla\cdot u=0.$$

For any $\alpha\in[0,1/2)$, consider initial data $u_0\in H^\alpha\cap L^{3,\infty}$ and solutions in $C([0,1]:H^\alpha_x)\cap L^2_tH^{\alpha+1}_x\cap L^\infty_tL^{3,\infty}_x$. Non-uniqueness conjecture. There are two different solutions $u_1,u_2$ with the same initial data $u_1(0)=u_2(0)=u_0$, both smooth in $(0,1]\times\mathbb{R}^3$, satisfying $t^{1/2}\|u_j(t)\|_{L^\infty}\lesssim 1$ for $j\in\{1,2\}$, and obeying

$$\frac{1}{2}\|u_j(t_0)\|_{L^2}^2=\frac{1}{2}\|u_j(t_1)\|_{L^2}^2+\int_{t_0}^{t_1}\|\nabla u_j(s)\|_{L^2}^2\,ds$$

\nfor every $t_0\leq t_1\in[0,1]$ and $j\in\{1,2\}$. In the strong form, the two solutions are axially symmetric and swirl-free. Establishing this would provide non-uniqueness of Leray-type solutions from singular initial data while retaining strong regularity, the energy identity, and axial symmetry without swirl; the supplied text presents it as the paper's main conjecture and gives numerical evidence and a theoretical framework rather than a resolution.