A preprint gives a constructive mathematical route to weak solutions of the three-dimensional incompressible Navier–Stokes equations that can come arbitrarily close to two prescribed endpoint states. Here, “weak solution” refers to a solution in the distributional sense used by the proof. The same framework also produces at least two different weak solutions from the same initial data for a dense subset of the finite-energy starting states.
The findings concern the flexibility and nonuniqueness of a mathematical model, not measurements from a laboratory or a study of human, animal or other physical samples. The document studies finite-energy mathematical vector fields and weak solutions on the three-dimensional torus, a periodic setting in which the construction relies on recurring trajectories.
The document is arXiv version 1, dated 20 August 2026, and is therefore a preprint. The supplied publication information does not report a journal, DOI or peer-review status.
What the construction shows
The central endpoint result is an approximation statement. Given any two admissible endpoint fields and any positive tolerance, the theorem constructs a weak solution whose starting and ending traces are each within that tolerance of the selected fields. The solution retains the uniform-in-time velocity and gradient regularity stated in the theorem.
In other words, the construction makes a dense collection of endpoint pairs attainable in the product energy topology. “Dense” means that, around any admissible pair, the method can find an attainable pair as close as desired. It does not mean that every requested pair is reached exactly.
The paper also states an exact nonuniqueness result, but with a specific scope. For a dense set of finite-energy initial data, there are at least two distinct weak solutions. Those solutions may agree throughout a nontrivial initial interval and then have different terminal traces.
The authors describe these conclusions as endpoint-trace density and exact nonuniqueness on a dense subset of finite-energy data. They distinguish both claims from exact controllability of arbitrary endpoint pairs and from a result covering every possible initial datum.
A moving-vortex construction
The proof uses convex integration, a deterministic iteration built from successive corrections to an approximate flow. Its building blocks are localized and rescaled Hill spherical vortices, combined with averaging along long orbits, an auxiliary source correction and a temporal corrector.
The argument tracks Reynolds stresses and uses stress decomposition as part of the iteration. These analytic devices allow the moving vortices and their corrections to be arranged so that the approximate equations can be driven toward a weak solution.
The Hill vortices are scaled so that the energy and the L6/5 gradient scales used by the argument are preserved. That scaling is central to carrying the corrections through the iteration while retaining the estimates required for the final weak solution.
The paper gives one compatible strict choice of iteration parameters: σ = 200, n = 20, γ = 10−3, μ = 6.002, η = 3 and p̄ = 6/5 + 5 × 10−5. These values are part of the proof’s parameter selection, rather than measurements from a dataset.
A compactness argument then turns the sequence of approximate Reynolds flows into a limiting weak solution. The paper reports strong convergence of the iterates and says the limit has the claimed gradient regularity uniformly in time.
Where the claim stops
The distinction between approximation and exact control is important. The result allows the endpoint errors to be made arbitrarily small, but it does not establish exact attainment of every prescribed pair of endpoint fields. Nor does it claim that the regularity exponent in the theorem is the best one possible.
The nonuniqueness statement is also narrower than a claim that the equation has multiple solutions from every finite-energy starting state. It applies to a dense subset, and the constructed solutions are not claimed to belong to the Leray–Hopf class or to satisfy its energy inequality.
The paper illustrates the energy issue with endpoint targets that begin at u(0) = 0 and end at a nonzero u(1). For 0 < ε < ∥u(1)∥L2/3, the constructed solution has an initial norm below ε and a terminal norm above 2ε. That behavior is incompatible with a nonincreasing kinetic-energy inequality.
The paper also does not assert L2-in-time H1 regularity or a local energy inequality for these solutions. Those omissions matter because the theorem is formulated in a finite-energy distributional class, not as a result about every stronger class of weak solutions.
The paper’s explicit regularity ceiling is presented as a limitation of this single-core construction, not as an impossibility theorem for every approach based on Hill vortices. It leaves open whether different localization or averaging mechanisms could improve the gain above the stated Hill threshold.
The argument is built on the three-dimensional torus and relies on periodic orbit recurrence. The supplied analysis does not establish that the same construction works on nonperiodic spaces, on bounded domains or on R3.
Questions left open
The work leaves several mathematical questions unresolved: what is the largest exponent compatible with endpoint-trace density and uniform-in-time gradient control; whether prescribed energy profiles can be combined with endpoint flexibility in Leray–Hopf or suitable classes; and whether other coherent vortices or domain-compatible constructions can extend the result.
Because this is a constructive analytic proof, its convergence is established through estimates and compactness rather than empirical validation. The result is therefore relevant to the mathematical study of weak-solution flexibility, convex integration and Navier–Stokes nonuniqueness, not direct physical or engineering evidence.
Disclosures
The acknowledgment reports support for Quoc-Hung Nguyen from the CAS Project for Young Scientists in Basic Research and the National Natural Science Foundation of China, including the grant identifiers listed in the paper.
The authors disclose that ChatGPT 5.6 assisted with language, organization, bibliographic and LaTeX tasks, as well as calculation verification. They state that it was not an independent source of the mathematical results, that they reviewed the AI-assisted revisions and that they take full responsibility for the paper’s contents.
Paper data and sources
Original title: Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices
Authors: Quoc-Hung Nguyen, Zexi Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text