Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices

We construct weak solutions of the three-dimensional incompressible Navier–Stokes equations on the torus. The convex-integration scheme is based on the moving-dipole construction of Bruè, Colombo, and Kumar~\cite{BrueColomboKumar2024}. For the explicit exponent $\bar{p}=\frac{6}{5}+5\times {10}^{-5},$ and for any two mean-zero, divergence-free vector fields in $L^2({\mathbb{T}}^3)$, we construct a weak solution whose traces at times $0$ and $1$ approximate the prescribed fields arbitrarily well and which satisfies $$u\in C([0,1];L^2({\mathbb{T}}^3)),\mathrm{\nabla }u\in C([0,1];L^{\bar{p}}({\mathbb{T}}^3))\mathrm{.}$$ Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in $L_{\sigma }^2({\mathbb{T}}^3)$.The principal perturbations are localized, rescaled copies of Hill’s spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the $L^{6\mathrm{/}5}$-scale of the velocity gradient. The construction uses localization of the potential exterior flow, long-orbit averaging of moving vortex cores, an auxiliary source correction, and a temporal corrector compatible with uniform-in-time Sobolev control.