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

Abstract

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 $\bar p=\frac65+5\times10^{-5},$ 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 \[ u\in C([0,1];L^2(\mathbb T^3)), \qquad \nabla u\in C([0,1];L^{\bar p}(\mathbb T^3)). \] Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in \(L^2_σ(\mathbb T^3)\).The principal perturbations are localized, rescaled copies of Hill's spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the \(L^{6/5}\)-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.

Disclosure

“dation of China under Grant Nos. 1251101538 and 12595282. He is grateful to Elia Bruè for suggesting that he consider this problem. AI use disclosure This paper was written by the authors and was not generated by artificial intelligence. OpenAI’s ChatGPT 5.6 was used to assist with language editing, organization, bibliographic checks, LATEX troubleshooting, and verification of mathematical calculations. It was not used as an independent source of mathematical results. All mathematical idea”

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2026-08-22T03:12:28.430283Z

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