Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

Exit data and the surviving quadratic velocity

The shrinking layer: integrate the normal flux first › Exit data and the surviving quadratic velocity · lines 894–924

The quadratic term in $v_{2}$ must not be halved. For a direct audit, the expanded second source contains both

- σ_{ε}^{'} a^{00} \partial_{t} w_{ε}^{ρ} and - σ_{ε} a^{00} \partial_{t}^{2} w_{ε}^{ρ} .

The first equation implies

\partial_{t}^{2} w_{ε}^{ρ} (ε s) = - ε^{- 1} χ^{'} (s) a_{Σ}^{0 ρ} + O (1) .

Consequently the singular part of the second source is

ε^{- 1} [- χ^{'} (s) A_{Σ}^{0 ρ} + 2 χ (s) χ^{'} (s) a_{Σ}^{00} a_{Σ}^{0 ρ}] .

Since

\int_{0}^{1} χ^{'} d s = 1, \int_{0}^{1} χ χ^{'} d s = \frac{1}{2},

the two quadratic contributions sum to the full term in (eq:velocities). This audit is consistent with, but not a substitute for, the integrated flux estimate.

Displayed formulas in this passage

unlabeled · line 899
$- σ_{ε}^{'} a^{00} \partial_{t} w_{ε}^{ρ} and - σ_{ε} a^{00} \partial_{t}^{2} w_{ε}^{ρ} .$
unlabeled · line 905
$\partial_{t}^{2} w_{ε}^{ρ} (ε s) = - ε^{- 1} χ^{'} (s) a_{Σ}^{0 ρ} + O (1) .$
unlabeled · line 910
$ε^{- 1} [- χ^{'} (s) A_{Σ}^{0 ρ} + 2 χ (s) χ^{'} (s) a_{Σ}^{00} a_{Σ}^{0 ρ}] .$
unlabeled · line 916
$\int_{0}^{1} χ^{'} d s = 1, \int_{0}^{1} χ χ^{'} d s = \frac{1}{2},$

Exit data and the surviving quadratic velocity

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