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

The two coordinate fluxes

The shrinking layer: integrate the normal flux first › The two coordinate fluxes · lines 738–765

X-D-ST@1#eq:fluxes

For $ρ = 0, 1, 2, 3$ , 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} w_{ε}^{ρ} + a^{0 i} \partial_{i} w_{ε}^{ρ}), = σ_{ε} \partial_{i} (A^{i ρ} + a^{i 0} \partial_{t} w_{ε}^{ρ} + a^{ij} \partial_{j} w_{ε}^{ρ}) . H_{1, ε}^{ρ} = σ_{ε} \partial_{i} a^{i ρ},

Spatial indices run from $1$ to $3$ . Both equations (eq:coordinates) have the form in Lemma lem:flux, and each $F_{j, ε} (0)$ vanishes.

Displayed formulas in this passage

X-D-ST@1#eq:fluxes · line 741
$F_{1, ε}^{ρ} F_{2, ε}^{ρ} H_{2, ε}^{ρ} = σ_{ε} a^{0 ρ}, = σ_{ε} (A^{0 ρ} + a^{00} \partial_{t} w_{ε}^{ρ} + a^{0 i} \partial_{i} w_{ε}^{ρ}), = σ_{ε} \partial_{i} (A^{i ρ} + a^{i 0} \partial_{t} w_{ε}^{ρ} + a^{ij} \partial_{j} w_{ε}^{ρ}) . H_{1, ε}^{ρ} = σ_{ε} \partial_{i} a^{i ρ},$