ns/bands — the ledger of scales

Near the singularity the construction is organised in dyadic bands where the concentration scale q is comparable to a fixed Q = 2^−ℓ. Within a band every quantity is rescaled so the collapsing annulus has size one, and three numbers run the show: the small parameter ε = Q^h, the slow scale S* = ℓ², and the carrier frequency k = ⌈ε^−1/2⌉ (6.1), (7.2). Every "smaller by a positive power of q" in the paper is a row of this table.

exponent h

first band shown · ℓ

The proof needs ε = q^h small, with h < 1/100. That is a very slow function. Pick how small you want ε and the page tells you how close to the singular time you have to be — and how that compares with anything a computer could resolve.

exponent h

target ε = q^h

q^h for the chosen h q^h at the proof's ceiling h = 1/100 double-precision spacing near 1, 2⁻⁵²

After the background and the leading pulses, what remains is corrected in a fixed four-operation cycle, and each cycle buys a fixed improvement in the residual's decay: σ_j = ⅕ + j/10 (9.8). The cycles are summed with cutoffs that switch each one on only very close to the singularity, so that on any fixed compact set only finitely many are active, while at q = 0 all of them are — and the residual becomes flat.

exponent h

stages to show

the four operations (Prop. 9.6)

1 · solve the sourced wave-amplitude equations for the nonzero angular harmonics (Prop. 9.1)
2 · signed amplitude increments whose cross-covariance with the leading pulses supplies the prescribed stress change (Prop. 7.6)
3 · correct the angular-mean flow by inverting the fast auxiliary-time derivative; rebuild the pressure (8.14), (8.20)
4 · solve the five radial moment equations (8.25)

A residual is flat if every Cartesian space–time derivative is O(q^N) for every N (§3.2). Flat functions are the ones that can be extended by zero and stay smooth. The proof never produces a closed form for its flat remainder; it proves the bound for each N separately, with constants that depend on N. Here is what that looks like against the textbook flat function e^−1/q.

compare at q

q^N, N = 1 … 20 e^−1/q, flat at 0 the paper's (3.4): |∂^α_x∂^b_tR| ≤ C_α,b,N q^N for every N

§3.4 · §3.6 · §6.1–6.2 · §9.3–9.5 · (6.1), (7.2), (9.8), (9.18) of the paper

Charts

For a dyadic index ℓ, Q = 2^−ℓ, ε = Q^h, S* = ℓ², and the chart coordinates are R = r/√Q, Z = z/Q^D, T = τ/Q. Velocity, pressure and residual are normalised by Q^A, Q^2A, Q^2A+½. In these units the concentrating annulus has bounded size, the viscous term carries the factor ε, and the fast auxiliary time advances at rate c_i ≍ 1/S*. Within a band ε and S* are constants; derivatives of any fixed order of the cutoffs cost powers of S*. The estimates the paper proves have the shape "≤ C ε^α S*^b" — a small parameter that is a power of Q, against a loss that is only polynomial in log(1/Q). Since S*^bε = ℓ^2b2^−hℓ → 0 for every b, the small parameter always wins eventually, and "eventually" is a lower bound ℓ ≥ ℓ₀, i.e. an upper bound q < q_big, that is fixed once and for all before the correction cycle starts (Lemma 9.7).

The rows of the ledger

ε = Q^h — the expansion parameter of the whole construction. Axial diffusion is weaker than radial by ε²; the background corrections come in powers ε²ⁿ (§5); the viscous term in the wave equation is O(ε) times a second derivative.
k = ⌈ε^−1/2⌉ — the carrier frequency, chosen so εk² ∈ [1, 4]: viscosity neither dominates nor disappears from the leading amplitude equation (7.2).
wavelength and amplitude — A_wave ≍ q^−½−h/2 and ℓ_wave ≍ q^½+h/2, so relative to the core's speed and radius both ratios are ≍ q^h/2 = √ε (§3.3). The pulses are small and short compared with the core, but only by √ε — barely.
A_wave² ≍ q^−1−h ≍ |u_θ|/q^½ — the squared amplitude is exactly the stress scale the annulus needs (§3.3).
S* = ℓ² — the slow scale: pulses last L_s ≍ S* in the fast time, slow cutoffs have mesh S*⁻³, and every estimate loses a power of it.
i(ℓ) = ⌊log_{T_g}(Q^−1−h/S*)⌋ — the covering index of the auxiliary torus for the band (6.5), with T_g = 4 + √2 the larger eigenvalue of the integer matrix J_g = [[3,1],[1,5]]. It is what makes the fast time advance at the right rate.
Re_θ ≍ q^−h = 1/ε — turns per diffusion time in the core.

How deep

With h < 1/100, ε = q^h falls below ½ only when q < 2⁻¹⁰⁰ ≈ 8·10⁻³¹. Double precision cannot represent a time 1 − q with q that small: the spacing of doubles near 1 is 2⁻⁵² ≈ 2·10⁻¹⁶. The construction is therefore not something a direct simulation could ever be expected to display; the regime in which its estimates hold lies far below any resolvable scale. That is not a flaw in the proof — asymptotic constructions are like this — but it is a fact worth having in front of you when reading claims about what the flow "looks like".

The correction cycle

Stage j holds fields (u^[j], p^[j]) whose supported residual is bounded, in chart units, by ε^σ_j times polynomial losses (9.18), with σ_j = ⅕ + j/10. One cycle — the four operations listed in the tab — produces stage j+1 with σ_j+1 = σ_j + 1/10. All stages live on one common domain (Lemma 9.7), and the loss of powers of q in derivative bounds depends on the derivative order but not on the stage (9.17). The stages are summed with cutoffs χ(a_jq), a_j+1 ≥ 2a_j, equal to one near q = 0 and supported in shrinking neighbourhoods; curls are taken after multiplying the vector potentials, so incompressibility survives exactly. The sum is locally finite for q > 0 and its residual, compared with any fixed finite stage, is flat (Proposition 9.9).