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

Lemma: Rescaled layer and endpoint data

The shrinking layer: integrate the normal flux first › Exit data and the surviving quadratic velocity · lines 842–894 · proof included or linked

X-D-ST@1#lem:profile, X-D-ST@1#eq:rescaled, X-D-ST@1#eq:velocities

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Uniformly for $0 \leq s \leq 1$ , in every local spatial smooth seminorm,

w_{ε}^{ρ} (ε s) \partial_{t} w_{ε}^{ρ} (ε s) v_{ε}^{ρ} (ε s) \partial_{t} v_{ε}^{ρ} (ε s) = - ε I_{1} (s) a_{Σ}^{0 ρ} + O (ε^{2}), = - χ (s) a_{Σ}^{0 ρ} + O (ε), = ε [- I_{1} (s) A_{Σ}^{0 ρ} + I_{2} (s) a_{Σ}^{00} a_{Σ}^{0 ρ}] + O (ε^{2}), = - χ (s) A_{Σ}^{0 ρ} + χ (s)^{2} a_{Σ}^{00} a_{Σ}^{0 ρ} + O (ε) .

The limiting positive-order initial positions are zero, and the exit velocities converge to

v_{1}^{ρ} = - a_{Σ}^{0 ρ}, v_{2}^{ρ} = - A_{Σ}^{0 ρ} + a_{Σ}^{00} a_{Σ}^{0 ρ} .

These limiting data depend on the profile endpoints, not its interior moments.

Proof. The fixed smooth coefficient difference $a (ε s) - a_{Σ}$ , and similarly for $A$ , is $O (ε)$ in every localized spatial norm. Use this in (eq:layer-bounds). The estimates in (eq:flux-bounds) also apply with the current $t$ in place of $ε$ ; their remainders are $O (t)$ . Integrating the velocities from zero therefore gives the position expansions with $O (ε^{2})$ remainders.

At $s = 1$ , $χ (1) = 1$ . The profile moments are multiplied by $ε$ in the positions, whereas the velocities approach (eq:velocities).

Displayed formulas in this passage

X-D-ST@1#eq:rescaled · line 846
$w_{ε}^{ρ} (ε s) \partial_{t} w_{ε}^{ρ} (ε s) v_{ε}^{ρ} (ε s) \partial_{t} v_{ε}^{ρ} (ε s) = - ε I_{1} (s) a_{Σ}^{0 ρ} + O (ε^{2}), = - χ (s) a_{Σ}^{0 ρ} + O (ε), = ε [- I_{1} (s) A_{Σ}^{0 ρ} + I_{2} (s) a_{Σ}^{00} a_{Σ}^{0 ρ}] + O (ε^{2}), = - χ (s) A_{Σ}^{0 ρ} + χ (s)^{2} a_{Σ}^{00} a_{Σ}^{0 ρ} + O (ε) .$
X-D-ST@1#eq:velocities · line 868
$v_{1}^{ρ} = - a_{Σ}^{0 ρ}, v_{2}^{ρ} = - A_{Σ}^{0 ρ} + a_{Σ}^{00} a_{Σ}^{0 ρ} .$

Lemma: Rescaled layer and endpoint data

Displayed formulas in this passage

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