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 805–833

The positive-order normal currents are

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

They obey conservation equations

\partial_{t} Π_{j, ε}^{ρ} = - \partial_{i} J_{j, ε}^{i ρ},

where

J_{1, ε}^{i ρ} J_{2, ε}^{i ρ} = - \partial_{i} w_{ε}^{ρ} + σ_{ε} a^{i ρ}, = - \partial_{i} v_{ε}^{ρ} + σ_{ε} (A^{i ρ} + a^{i ν} \partial_{ν} w_{ε}^{ρ}) .

Their layer integrals are $O (ε)$ . These are completion-normal currents, not host evolution equations.

Displayed formulas in this passage

unlabeled · line 821
$J_{1, ε}^{i ρ} J_{2, ε}^{i ρ} = - \partial_{i} w_{ε}^{ρ} + σ_{ε} a^{i ρ}, = - \partial_{i} v_{ε}^{ρ} + σ_{ε} (A^{i ρ} + a^{i ν} \partial_{ν} w_{ε}^{ρ}) .$
unlabeled · line 808
$Π_{1, ε}^{ρ} = \partial_{t} w_{ε}^{ρ} + F_{1, ε}^{ρ}, Π_{2, ε}^{ρ} = \partial_{t} v_{ε}^{ρ} + F_{2, ε}^{ρ} .$
unlabeled · line 816
$\partial_{t} Π_{j, ε}^{ρ} = - \partial_{i} J_{j, ε}^{i ρ},$