Manuscript X-D@3 · X-D-v3.tex · lemma · b7774529

Lemma: Rescaled layer, exit data, and profile independence

D2: normal-flux estimate and the profile lemma › Application to the two coordinate coefficients · lines 765–849 · proof included or linked

X-D@3#lem:profile, X-D@3#eq:rescaled, X-D@3#eq:velocities, X-D@3#eq:ramp-terms

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Write

a_{Σ} = a (0, \cdot), A_{Σ} = A (0, \cdot), I_{ℓ} (s) = \int_{0}^{s} χ (r)^{ℓ} d r, ℓ = 1, 2.

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

X_{1, ε}^{ρ} (ε s) \partial_{t} X_{1, ε}^{ρ} (ε s) X_{2, ε}^{ρ} (ε s) \partial_{t} X_{2, 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 exit positions tend to zero, and the exit velocities tend to

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

The limiting data depend only on the profile endpoints.

Proof. Replace the smooth coefficients at $ε s$ by their initial traces in (eq:first-layer)–(eq:second-layer). The error is $O (ε)$ . Integrating the velocities gives the position formulas.

To display every ramp-derivative contribution, the expanded sources are

□ X_{1, ε}^{ρ} = □ X_{2, ε}^{ρ} = - σ_{ε}^{'} a^{0 ρ} - σ_{ε} \partial_{t} a^{0 ρ} - σ_{ε} \partial_{i} a^{i ρ}, - σ_{ε}^{'} A^{0 ρ} - σ_{ε} \partial_{t} A^{0 ρ} - σ_{ε} \partial_{i} A^{i ρ} - σ_{ε}^{'} a^{0 ν} \partial_{ν} X_{1, ε}^{ρ} - σ_{ε} \partial_{t} (a^{0 ν} \partial_{ν} X_{1, ε}^{ρ}) - σ_{ε} \partial_{i} (a^{i ν} \partial_{ν} X_{1, ε}^{ρ}) .

The first equation implies

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

The singular part of the second source is therefore

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

One quadratic contribution comes from $- σ_{ε}^{'} a^{00} \partial_{t} X_{1, ε}$ ; the other comes from $- σ_{ε} a^{00} \partial_{t}^{2} X_{1, ε}$ . Their integrals each give half of the quadratic velocity:

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

Their sum agrees with the exact flux calculation. Discarding either term gives an incorrect exit velocity.

The moments $I_{1} (1), I_{2} (1)$ multiply $ε$ in the exit positions. Other profile-dependent moments similarly occur in remainders, not in (eq:velocities).

Displayed formulas in this passage

X-D@3#eq:rescaled · line 773
$X_{1, ε}^{ρ} (ε s) \partial_{t} X_{1, ε}^{ρ} (ε s) X_{2, ε}^{ρ} (ε s) \partial_{t} X_{2, ε}^{ρ} (ε s) = - ε I_{1} (s) a_{Σ}^{0 ρ} + O (ε^{2}), = - χ (s) a_{Σ}^{0 ρ} + O (ε), = ε [- I_{1} (s) A_{Σ}^{0 ρ} + I_{2} (s) a_{Σ}^{00} a_{Σ}^{0 ρ}] + O (ε^{2}), = - χ (s) A_{Σ}^{0 ρ} + χ (s)^{2} a_{Σ}^{00} a_{Σ}^{0 ρ} + O (ε) .$
X-D@3#eq:velocities · line 790
$v_{1}^{ρ} = - a_{Σ}^{0 ρ}, v_{2}^{ρ} = - A_{Σ}^{0 ρ} + a_{Σ}^{00} a_{Σ}^{0 ρ} .$
X-D@3#eq:ramp-terms · line 807
$□ X_{1, ε}^{ρ} = □ X_{2, ε}^{ρ} = - σ_{ε}^{'} a^{0 ρ} - σ_{ε} \partial_{t} a^{0 ρ} - σ_{ε} \partial_{i} a^{i ρ}, - σ_{ε}^{'} A^{0 ρ} - σ_{ε} \partial_{t} A^{0 ρ} - σ_{ε} \partial_{i} A^{i ρ} - σ_{ε}^{'} a^{0 ν} \partial_{ν} X_{1, ε}^{ρ} - σ_{ε} \partial_{t} (a^{0 ν} \partial_{ν} X_{1, ε}^{ρ}) - σ_{ε} \partial_{i} (a^{i ν} \partial_{ν} X_{1, ε}^{ρ}) .$
unlabeled · line 768
$a_{Σ} = a (0, \cdot), A_{Σ} = A (0, \cdot), I_{ℓ} (s) = \int_{0}^{s} χ (r)^{ℓ} d r, ℓ = 1, 2.$
unlabeled · line 824
$\partial_{t}^{2} X_{1, ε}^{ρ} (ε s) = - ε^{- 1} χ^{'} (s) a_{Σ}^{0 ρ} + O (1) .$
unlabeled · line 829
$ε^{- 1} [- χ^{'} (s) A_{Σ}^{0 ρ} + 2 χ (s) χ^{'} (s) a_{Σ}^{00} a_{Σ}^{0 ρ}] .$
unlabeled · line 839
$\int_{0}^{1} χ^{'} (s) d s = 1, \int_{0}^{1} χ (s) χ^{'} (s) d s = \frac{1}{2} .$

Lemma: Rescaled layer, exit data, and profile independence

Displayed formulas in this passage

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