Simplified vortex-core illustration (rescaled)
Fluid enters from the sides and leaves above and below the central layer.
Fluid flows outward along the axis.
Fluid circles the axis while moving inward.
As t → 1⁻
Peak speed becomes unbounded as the core shrinks to the origin.
The full, smoothly forced flow keeps total kinetic energy bounded.
Time remaining τ=1−t
Height / radius · relative to start
Characteristic speed
t→1−illustrativeh=0.0005
Initializing WebGPU.
A numerical illustration of the leading inner core, using illustrative parameters. This is not a numerical reproduction of the paper’s complete solution.
The announced Navier–Stokes result constructs flow from rest, driven by a smooth force compactly supported in space and time. Peak velocity becomes unbounded as t approaches 1 from below while total kinetic energy stays bounded. This animation starts much later, at a near-singularity reference snapshot. It evaluates a prescribed leading field; it does not evolve the complete forced equations from rest.
The story uses instantaneous streamlines and flow markers on a slowed display clock. Its meridional, equatorial and orthographic views share the same leading field and marker population.
The color key shows angular speed ω = uθ/r, normalized separately at each height between the radial boundary and axis. Azimuthal linear speed equals rω and vanishes on the axis. The colors do not measure total speed |u|, or compare absolute speeds across heights or times. Lighting and marker opacity affect brightness. The rotation profile that colors a point is the same one the streamline integration turns it with, so the colors and the marker motion describe one field.
The radius readout uses R/Rref = (τ/τref)1/2 at fixed similarity coordinates. Height shrinks slightly more slowly, with H/Href = (τ/τref)1/2−h; the concentration chapter shows the resulting change in height/radius. The two length ratios can round to the same displayed value early in the story, but they are not equal. The start means the displayed reference snapshot, not the initial state at rest. The camera removes overall axial contraction, and the expanded axial map improves visibility; the fixed x, y, z guides show directions, not equal physical units. Logarithmic readouts remain finite even when the length ratios are too small to represent directly.
Checked against the article, paper §2.1 and §3.1, and the Lean sources on 12 September 2026. The radius and velocity readouts are ratios of the prescribed scale functions. All ref subscripts refer to the first displayed snapshot. The changing numerical equations use ≈ because their exponents are rounded to two decimal places; the aspect multiplier is rounded to three. The definition τ = 1 − t and the chosen h remain exact. Time is normalized so the singular time is 1, approached from below. The energy symbol ≍ means bounds above and below by fixed positive multiples as τ → 0; it does not assert exact proportionality. Axial stretching describes local outflow, even while the core’s overall height shrinks. The full theorem also needs the exterior, oscillatory pulses, corrections and smooth force; this visual check does not certify those omitted parts or the illustrative parameter choices.
The renderer uses numerically solved profiles U(X, η) and ϕ(X, η) rather than the paper’s Bessel-type comparison series alone. The offline solve includes the pressure profile Π and targets the paper’s leading equations (4.7), (4.13) and (B.12)–(B.16). A shared streamfunction potential determines both meridional velocity components. Cubic interpolation keeps them consistent between table samples.
The chosen inputs are Λ = 64, C = 64, σ* = 0.5, j = 0.0005 and Π₀(η) = −(1 + η²)⁻², at viscosity ν = 1. These are illustrative axis data, not the pressure datum obtained by matching the paper’s exterior. The chosen σ* fails the shear-complement condition (B.2): at the relevant source zero, χ ≈ 4.89 × 10⁻⁸, where the construction requires χ > 0.99. The exponent h = 0.0005 also violates the outer schedule: Lemma 4.8 fixes Td = eMd + 10 and requires 0 < h < min{1/100, λ, e−Td}, so h < e−10 ≈ 4.5 × 10−5; (A.6) sharpens this to h ≪ e−Td, and §4.6 carries these choices into the leading profile. Those are sufficient conditions used by the proof, not a bound on every possible blowup exponent. These are limitations of this demo’s parameter choices, not findings against the paper or Lean proof. Reproducing that schedule requires a new, consistently chosen parameter set and new core profiles.
Both lengths shrink, and the radius shrinks faster than the height, so the core grows relatively more slender. The azimuthal and axial characteristic speeds scale as τ−1/2−h; radial speed has the different scale τ−1/2. These are normalized powers, not measured maxima. The core-energy power describes a shrinking similarity region; the theorem’s bounded total energy is a separate global result.
The reference angular factor is K = 12, corresponding to τref−h/C = K. Thus log₁₀ τref = −log₁₀(CK)/h: the opening snapshot is already in a near-singularity sequence, not the initial rest state. The scrubber measures −log₁₀(τ/τref). These are orders of magnitude in the remaining time, not elapsed years. At 1,200 the remaining time is τref × 10⁻¹²⁰⁰; the field is still at a finite time before t = 1. The blow-up corresponds to τ → 0 and an unbounded logarithmic coordinate, not the end of this chosen visualization range. At h = 0.0005, 1,200 orders of magnitude increase the relative aspect ratio by about four. The paper’s Figure 1 exaggerates that difference for illustration.
The expanded projection maps the reference axial coordinate zref to 0.72 asinh(zref / 0.003). It spreads the rapidly winding region near the central plane into a tall structure. This is a monotone coordinate transformation of the computed curves, not a change to the velocity equations. It changes local pitch and proportions. Incompressibility is assessed in the original physical coordinates.
The reference radial and axial units are independent, so the drawing is not in equal physical units. The concentration powers are applied after projection, and each camera applies uniform zoom, which preserves the physical aspect ratio it shows. The readouts describe the original field’s scaling laws, not literal distances or speeds in the expanded drawing.
The 66-second story shows one instantaneous leading-core field through a moving orthographic camera and three persistent side views. Every ribbon and flow marker comes from the same time-parametrized streamline buffer, with additional azimuthal seeds allowed by axisymmetry. These markers show motion through each frozen field; they are not material histories through the evolving solution.
During playback, the current slab opens into the full three-dimensional particle volume before each camera turn. The camera then rotates through that unchanged coordinate space; the next slab fades in only after the turn. Uniform camera zoom pulls back before leaving the top view and closes in after approaching it. Selecting a view starts the camera and representation changes immediately and settles within 0.8 seconds. The main scene retains the same lit ribbons throughout, with a constant base marker opacity. Slab selection changes the ribbons’ lighting and width while retaining the surrounding blue structure as context. Only the moving particles are slab-filtered; dimmer surrounding ribbons remain part of the same field.
The x, y and z guides show the mapped 3D coordinate directions. The top-down x and y guides shorten smoothly with the lens scale to keep the circle readable. The axis letters are font-outline meshes, positioned in 3D and depth-tested against the ribbons and visible particle cores. Lines and letters share a continuous fade as their axes turn end-on. The guides indicate orientation, without physical-unit tick marks. Eleven mechanics explanations occupy six-second story beats, with brief fades at text changes, connected to projected locations in the leading field. A curved arrow around the equatorial slab shows the sense of rotation; it does not represent a transport or a force. During the two contraction beats, a radial guide follows a fixed similarity radius just below the central layer. These overlays are orientation and explanation guides, not additional simulated particles or force measurements. The meridional slab keeps |yref| below about 0.035 and the equatorial slab keeps |zref| below about 0.015, with soft edges, in the reference coordinates before the axial map and contraction scales. The drawn streamlines start inside rref ≈ 0.35 and reach |zref| ≈ 0.6, so each slab is a thin layer: ribbons outside it are dimmed and markers outside it are hidden. The equatorial view looks along the axis at that thin slab of markers, with the full ribbon structure dimmed behind it. Side and 3D cameras follow the height; the top camera follows the radius. Their separate uniform magnifications keep each view visible; displayed sizes across panels are not physical size comparisons. The physical radius and height both shrink.
All highlights share a deterministic, slowed display clock: playback and scrubbing reproduce the same phases. The normalized streamline clock accelerates continuously from 0.12 to 0.48 units per presentation second: a compressed fourfold display ramp over the whole story, without chapter-dependent jumps. Its analytic integral makes scrubbing deterministic. Angular winding still changes with the evaluated field; marker speed is not a physical speed measurement. The velocity-scale readout uses V/Vref = (τ/τref)^(−½−h), independently of that slowed display clock; at h = 0.0005 and 1,200 orders of remaining time it reads 10^600.60. The large physical speed increase cannot be played literally at a finite frame rate. Short trails follow the curved streamline in up to four WebGPU segments, with bounded angular exposure to preserve the direction of fast motion. Markers fade before being recycled. The entire timeline uses one logarithmic mapping: at progress p, τ/τref = 10^(−1200p). Equal distances on the scrubber multiply the time remaining by equal factors; halfway is 10^−600. The ticks show τ/τref, with intermediate ticks omitted on narrow screens. The 0:00–1:06 counter measures presentation time. The ending clears the captions and their WebGPU leaders before revealing the limiting statement over the dimmed preview; it does not draw a computed state at the singular time. The final equation gives the paper’s asymptotic velocity and core-energy scales, not a computed global energy integral or completed smooth-force construction.
No empty slot is part of this fluid model. Section 2.1 and Figure 1 describe a dividing layer near z = 0: axial motion has opposite directions on its two sides, while fluid continues spiralling inward. The sampling reaches close to that layer. Individual strands are selected streamlines, not the boundary of the fluid.
Numerical checks cover the leading equations, the shipped profile table, interpolation, incompressibility, streamline direction, time scaling and GPU results. The leading equations agree in sampled numerical checks, including inward central flow, opposite axial outflows, centrifugal-pressure balance and incompressibility. The radial and axial contraction powers and snapshot ratios are consistent with the evaluated field. The finite interpolant is not the paper’s smooth analytic field.
Lean’s profile equations, heat profile, outgoing pressure and correction construction were reviewed at 8937a8f; the relevant statements and equations were rechecked against commit f9e8bc5 on 12 September 2026. Its existence proofs select parameters, inverses and correction thresholds; they do not supply a ready-to-render numerical dataset. The Lean project was not independently built here, and this JavaScript/WGSL translation is not formally verified.
8937a8f
f9e8bc5
The full result needs cancellation of the complete momentum residual so that the force and all its derivatives remain smooth. Leading balance alone does not establish this. The initial rest state, matched exterior and infinite correction construction are omitted from the animation. No infinite velocity is drawn: the sequence ends at a finite time before t = 1.