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 $\bar p=\frac65+5\times10<sup>{-5},$ and for any two mean-zero, divergence-free vector fields in $L^2(\mathbb T<sup>3)$, 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.

• The paper constructs weak 3D Navier–Stokes solutions connecting arbitrary mean-zero divergence-free endpoint data within any prescribed L² error, with ∇u in C([0,1];L^{6/5+5×10⁻⁵}).
• The moving Hill-vortex convex-integration scheme uses localized, rescaled vortices on rational trajectories, temporal scheduling, orbit averaging, and correctors to overcome viscous and transport errors while preserving endpoint control.
• The construction yields exact nonuniqueness for a dense set of finite-energy initial data, but its solutions generally violate the Leray–Hopf energy inequality and remain far below the Navier–Stokes uniqueness threshold near p=3/2.

This paper by Nguyen and Wang constructs weak solutions of the three-dimensional incompressible Navier–Stokes equations on $\mathbb T^3$ with prescribed endpoint traces, using a convex-integration scheme whose principal perturbations are localized, rescaled copies of Hill's spherical vortex moving along rational trajectories. The work adapts to the viscous three-dimensional setting the moving-dipole strategy of Brué, Colombo, and Kumar for two-dimensional Euler (Bruè et al., 2024), and it yields both an approximate controllability statement and exact nonuniqueness on a dense set of finite-energy data.

Main results

Fix $\nu>0$ and consider the unforced Navier–Stokes equations on $\mathbb T^3$. The central theorem states that for the explicit exponent

$\bar p = \frac65 + 5\times 10^{-5},$

and for any two mean-zero divergence-free fields $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$ and any $\varepsilon>0$, there exists a weak solution $u\in C([0,1];L^2(\mathbb T^3))$ with $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$ such that

$$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$

Equivalently, the set of endpoint pairs of weak solutions in this class is dense in $L^2_\sigma\times L^2_\sigma$: no compatibility condition between the two targets is required. This is an approximate controllability statement; neither trace is attained exactly.

Two consequences deserve emphasis. First, exploiting the time-locality of the iteration, the authors obtain exact nonuniqueness: there exists a dense set $\nu>0$0 such that every datum in $\nu>0$1 launches at least two distinct weak solutions that agree identically on $\nu>0$2 for some $\nu>0$3 but have different terminal traces. Second, the flexibility is incompatible with the Leray–Hopf energy inequality: taking $\nu>0$4, $\nu>0$5, and $\nu>0$6, the constructed solution has strictly increasing kinetic energy and therefore lies outside the Leray–Hopf class. A modest fixed-time spatial gain follows from Sobolev embedding: since $\nu>0$7, one has $\nu>0$8, though the class remains far below the Ladyzhenskaya–Prodi–Serrin uniqueness range.

The authors are explicit about what is not claimed. The exponent $\nu>0$9 is not asserted optimal; the solutions do not satisfy the energy inequality; nonuniqueness holds only on a dense subset, not all of $\mathbb T^3$0.

Hill vortices and the critical scaling

Hill's spherical vortex is an axisymmetric, no-swirl traveling Euler solution with azimuthal vorticity supported in a ball. After subtracting the far-field velocity, the decaying profile $\mathbb T^3$1 satisfies $\mathbb T^3$2, and outside its core coincides with the potential doublet $\mathbb T^3$3. This potential character permits localization by cutting off the exterior field without altering the vorticity core.

The rescaled profile uses core radius $\mathbb T^3$4 and amplitude $\mathbb T^3$5:

$\mathbb T^3$6

which gives $\mathbb T^3$7 and $\mathbb T^3$8. Consequently the kinetic energy and the $\mathbb T^3$9-gradient norm are invariant under this scaling — in agreement with $\bar p = \frac65 + 5\times 10^{-5},$0 — while for $\bar p = \frac65 + 5\times 10^{-5},$1 concentration creates a loss $\bar p = \frac65 + 5\times 10^{-5},$2 with $\bar p = \frac65 + 5\times 10^{-5},$3. This identifies $\bar p = \frac65 + 5\times 10^{-5},$4 as the Hill threshold. Notably, this threshold differs from the Navier–Stokes scale-invariant exponent $\bar p = \frac65 + 5\times 10^{-5},$5 (under $\bar p = \frac65 + 5\times 10^{-5},$6); the construction crosses the Hill threshold but remains supercritical for Navier–Stokes.

A Helmholtz correction restores incompressibility after localization, and choosing the intrinsic trajectory $\bar p = \frac65 + 5\times 10^{-5},$7 cancels the leading transport term against the quadratic Euler self-interaction. What remains is a distinguished vector source proportional to $\bar p = \frac65 + 5\times 10^{-5},$8, whose spatial mean equals the time derivative of the block's impulse $\bar p = \frac65 + 5\times 10^{-5},$9; viscosity contributes a symmetric stress estimated at an auxiliary exponent $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$0 where the scaling exponent is positive.

The iteration scheme

The scheme iterates the Navier–Stokes–Reynolds system, in which the defect of stage $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$1 appears as a symmetric stress $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$2. At each stage:

• Mollification regularizes the background flow at scale $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$3.
• Geometric decomposition resolves the mollified stress into nine positive rank-one pieces along rationally oriented directions $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$4 whose orbits have periods comparable to $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$5 and are $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$6-dense. A perturbation matrix $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$7 with rational component ratios ensures these properties without parity assumptions.
• Time partition divides each cell into nine disjoint subintervals, activating at most one direction at a time; coefficients are frozen at their temporal averages, and disjoint scheduling removes leading cross-direction interactions.
• Moving Hill blocks treat each frozen coefficient $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$8: the radius is coupled to the coefficient via $u^{(0)}, u^{(1)}\in L^2_\sigma(\mathbb T^3)$9, the amplitude is normalized so that $\varepsilon>0$0, and the vortex speed is inversely proportional to $\varepsilon>0$1, so the core spends more residence time where the stress is larger.
• Orbit averaging converts the concentrated source into a directional tensor flux: integration by parts along the closed rational orbit recovers $\varepsilon>0$2 up to an error $\varepsilon>0$3 of size $\varepsilon>0$4.
• Temporal corrector $\varepsilon>0$5, defined as the Leray projection of the primitive of the zero-cell-average source, restores pointwise-in-time cancellation, vanishes at cell interfaces (hence at $\varepsilon>0$6), and gains the short factor $\varepsilon>0$7.

Because blocks and corrector vanish at cell interfaces, the endpoint errors satisfy $\varepsilon>0$8, which is summable; combined with smooth initialization interpolating the two mollified targets, this gives the endpoint approximation.

Parameter balance above the threshold

The central analytical difficulty is that concentration has opposite effects: small radii improve the intrinsic and viscous errors below $\varepsilon>0$9 but increase the gradient norm above it. For a principal block of amplitude $u\in C([0,1];L^2(\mathbb T^3))$0 and smallest radius $u\in C([0,1];L^2(\mathbb T^3))$1,

$u\in C([0,1];L^2(\mathbb T^3))$2

so summability requires $u\in C([0,1];L^2(\mathbb T^3))$3. Orbit completion and temporal-corrector estimates impose further constraints linking $u\in C([0,1];L^2(\mathbb T^3))$4, $u\in C([0,1];L^2(\mathbb T^3))$5, $u\in C([0,1];L^2(\mathbb T^3))$6, $u\in C([0,1];L^2(\mathbb T^3))$7, and $u\in C([0,1];L^2(\mathbb T^3))$8. The authors exhibit the compatible choice

$u\in C([0,1];L^2(\mathbb T^3))$9

with intrinsic exponents $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$0, $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$1, $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$2, and source exponent $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$3, verifying every inequality strictly; for $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$4 the critical left-hand side is approximately $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$5. The narrowness of this margin explains why the gain above $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$6 is small, and the authors note that temporal concentration cannot substitute for this balance because it improves only time-integrated norms, not the required $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$7 bound.

Nonuniqueness via time locality

Density of endpoint pairs alone does not imply same-data nonuniqueness. The additional ingredient is a coupling lemma: if two Reynolds triples agree up to a time $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$8, then one step of the iteration can be performed identically on all complete cells contained in $\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3))$9, since mollification, freezing, trajectories, sources, and the cellwise corrector depend only on the past. Running two constructions from initial data agreeing near $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$0 (both built from a common mollified field $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$1) toward distinct targets $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$2 and $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$3 of unit norm, with $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$4, produces two limiting solutions agreeing on $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$5 with terminal traces separated by more than $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$6 in $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$7. Since the common initial trace can be made arbitrarily close to any prescribed $$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$8, the set of data admitting multiple solutions is dense.

Limitations and open questions

The paper devotes a section to the limits of the single-core construction. Under the present parameter regime, the formal ceiling imposed by the principal-gradient estimate is

$$\|u(\cdot,0)-u^{(0)}\|_{L^2}<\varepsilon,\qquad \|u(\cdot,1)-u^{(1)}\|_{L^2}<\varepsilon.$$9

so the achieved $L^2_\sigma\times L^2_\sigma$0 sits essentially at this ceiling; reoptimizing $L^2_\sigma\times L^2_\sigma$1 and $L^2_\sigma\times L^2_\sigma$2 may improve the margin slightly but does not approach $L^2_\sigma\times L^2_\sigma$3, since increasing $L^2_\sigma\times L^2_\sigma$4 and decreasing $L^2_\sigma\times L^2_\sigma$5 play competing roles and the orbit-spacing condition keeps $L^2_\sigma\times L^2_\sigma$6 large. The authors also show quantitatively that temporal concentration worsens the fixed-time gradient estimate, that anisotropic cutoffs leave the intrinsic Hill-core loss unchanged, and that splitting one directional stress among $L^2_\sigma\times L^2_\sigma$7 parallel cores introduces a factor $L^2_\sigma\times L^2_\sigma$8 that grows for $L^2_\sigma\times L^2_\sigma$9 unless packing improves. Open questions include: whether optimization or a different averaging mechanism yields a larger uniform-in-time gain; whether energy profiles can be prescribed within this class; whether alternative coherent structures (Fraenkel rings, Norbury's family) with better energy–gradient scaling can meet the requirements of exact traveling cancellation, controllable impulse, localizability, and small viscous stress; and whether the construction extends to $\nu>0$00 or bounded domains, where periodic recurrence is unavailable.

Conclusion

The paper establishes density of attainable endpoint traces for finite-energy weak solutions of the three-dimensional Navier–Stokes equations with $\nu>0$01, $\nu>0$02, together with exact nonuniqueness on a dense set of initial data. Its technical contribution is the identification of the Hill-vortex scaling as the obstruction at $\nu>0$03 and the explicit resolution of the competing concentration, amplitude, and orbit-averaging constraints that permit a strict crossing of that threshold. The solutions lie below the Leray–Hopf class, and the method's ceiling near $\nu>0$04 leaves open whether structural modifications of the building block can substantially raise the attainable regularity.