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

Proposition: Uniform layer control

The shrinking layer: integrate the normal flux first › The two coordinate fluxes · lines 766–805 · proof included or linked

X-D-ST@1#prop:layer, X-D-ST@1#eq:layer-bounds

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For every spatial Sobolev order, uniformly on $0 \leq t \leq ε$ ,

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

The corresponding local spatial smooth estimates also hold.

Proof. The first flux pair is bounded in every spatial Sobolev norm independently of $ε$ . Lemma lem:flux therefore gives $w_{ε} = O (ε)$ and $\partial_{t} w_{ε} = - σ_{ε} a^{0 \cdot} + O (ε)$ .

These bounds, in sufficiently high spatial orders, make both $F_{2, ε}$ and $H_{2, ε}$ uniformly bounded in any prescribed spatial Sobolev order. In particular, no estimate of $\partial_{t}^{2} w_{ε}$ is needed to establish their boundedness.

Applying the lemma again gives $v_{ε} = O (ε)$ and $\partial_{t} v_{ε} = - F_{2, ε} + O (ε)$ . Substituting the first-coordinate estimates in (eq:fluxes) gives the final expression. Sobolev embedding gives the local spatial smooth version.

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X-D-ST@1#eq:layer-bounds · line 770
$w_{ε} v_{ε} = O (ε), = O (ε), \partial_{t} w_{ε} \partial_{t} v_{ε} = - σ_{ε} a^{0 \cdot} + O (ε), = - σ_{ε} A^{0 \cdot} + σ_{ε}^{2} a^{00} a^{0 \cdot} + O (ε) .$

Proposition: Uniform layer control

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