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

Application to the two coordinate coefficients

D2: normal-flux estimate and the profile lemma › Application to the two coordinate coefficients · lines 707–764

X-D@3#eq:fluxes, X-D@3#eq:first-layer, X-D@3#eq:second-layer

Define

F_{1, ε}^{ρ} F_{2, ε}^{ρ} H_{2, ε}^{ρ} = σ_{ε} a^{0 ρ}, = σ_{ε} (A^{0 ρ} + a^{00} \partial_{t} X_{1, ε}^{ρ} + a^{0 i} \partial_{i} X_{1, ε}^{ρ}), = σ_{ε} \partial_{i} (A^{i ρ} + a^{i 0} \partial_{t} X_{1, ε}^{ρ} + a^{ij} \partial_{j} X_{1, ε}^{ρ}) . H_{1, ε}^{ρ} = σ_{ε} \partial_{i} a^{i ρ},

Spatial indices range from $1$ to $3$ . Both normal flux terms vanish initially, and

□ X_{j, ε}^{ρ} = - \partial_{t} F_{j, ε}^{ρ} - H_{j, ε}^{ρ} .

The first application of Lemma lem:flux gives

X_{1, ε} = O (ε), \partial_{t} X_{1, ε} = - σ_{ε} a^{0 \cdot} + O (ε) .

Thus $F_{2, ε}, H_{2, ε}$ are uniformly bounded in every spatial seminorm, without a bound on $\partial_{t}^{2} X_{1, ε}$ . The second application gives

X_{2, ε} = O (ε), \partial_{t} X_{2, ε} = - σ_{ε} A^{0 \cdot} + σ_{ε}^{2} a^{00} a^{0 \cdot} + O (ε) .

These are also exact normal-current estimates. For $X_{ε} = x + κ X_{1, ε} + κ^{2} X_{2, ε}$ , the positive-order normal currents are

Π_{1, ε}^{ρ} = \partial_{t} X_{1, ε}^{ρ} + F_{1, ε}^{ρ}, Π_{2, ε}^{ρ} = \partial_{t} X_{2, ε}^{ρ} + F_{2, ε}^{ρ} .

They satisfy $\partial_{t} Π_{j, ε} = - \partial_{i} J_{j, ε}^{i}$ and start at zero. The spatial currents are uniformly bounded, so their layer integrals are $O (ε)$ .

Displayed formulas in this passage

X-D@3#eq:fluxes · line 710
$F_{1, ε}^{ρ} F_{2, ε}^{ρ} H_{2, ε}^{ρ} = σ_{ε} a^{0 ρ}, = σ_{ε} (A^{0 ρ} + a^{00} \partial_{t} X_{1, ε}^{ρ} + a^{0 i} \partial_{i} X_{1, ε}^{ρ}), = σ_{ε} \partial_{i} (A^{i ρ} + a^{i 0} \partial_{t} X_{1, ε}^{ρ} + a^{ij} \partial_{j} X_{1, ε}^{ρ}) . H_{1, ε}^{ρ} = σ_{ε} \partial_{i} a^{i ρ},$
X-D@3#eq:first-layer · line 735
$X_{1, ε} = O (ε), \partial_{t} X_{1, ε} = - σ_{ε} a^{0 \cdot} + O (ε) .$
X-D@3#eq:second-layer · line 743
$X_{2, ε} = O (ε), \partial_{t} X_{2, ε} = - σ_{ε} A^{0 \cdot} + σ_{ε}^{2} a^{00} a^{0 \cdot} + O (ε) .$
unlabeled · line 729
$□ X_{j, ε}^{ρ} = - \partial_{t} F_{j, ε}^{ρ} - H_{j, ε}^{ρ} .$
unlabeled · line 755
$Π_{1, ε}^{ρ} = \partial_{t} X_{1, ε}^{ρ} + F_{1, ε}^{ρ}, Π_{2, ε}^{ρ} = \partial_{t} X_{2, ε}^{ρ} + F_{2, ε}^{ρ} .$

Application to the two coordinate coefficients

Displayed formulas in this passage

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