Evolution in the outer domains of Navier-Stokes could allow finite-time dissipation to form without singularities.

Could the enstrophy $Z$ scaling regime $\sqrt{\nu}Z$ originally identified during trefoil vortex knot reconnection (JFM 839, R2, 2018), be universal? And can it lead to finite energy dissipation $\Delta E_\epsilon=\int_0^{t_\epsilon} \epsilon\,dt$ as $\nu\to0$? This reconnection regime is characterized by linear $B_\nu(t)=(\sqrt{\nu}Z(t))^{-1/2}$ up to a time $t_x$ with fixed $B_\nu(t_x)$. Its enstrophy growth for $T_c(\nu)>t_x$ is $Z_{\nu}(t) \sim \nu^{-1/2}(T_c(\nu)-t)^{-2}$, giving an energy dissipation rate $\epsilon(t_x)=\nu Z(t_x)\to0$ as $\nu\to0$. Nested coiled rings also have this scaling as they reconnect. Recently, two anti-parallel vortex calculations have gotten $B_\nu(t)$ scaling during reconnection, identified by finite circulation exchange $\Delta\Gamma$. For trefoils, finite $\epsilon$ appears at $t_\epsilon\approx 2t_x$, with similar results for very long anti-parallel vortices. By taking advantage of the anti-parallel symmetries, the new high-resolution data can identify a front perpendicular to the line of reconnection that would be blocked if the domains were fixed, perhaps explaining why the $B_\nu(t)$ scaling requires growing domains. These finite-time events are consistent Sobolev regularity inequalities, forming finite $\Delta E$ without singularities.