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

Proposition: The general rate is not o(ε)

Diagnostics and sharpness of the rate › Conformal temporal diagnostic · lines 1829–1867 · proof included or linked

X-D-ST@1#prop:optimal

There are fixed admissible data for which $d_{1, ε} - d_{1, 0}$ is not $o (ε)$ in $D_{C}$ , or after a fixed free-solution evaluation.

Proof. Use the conformal family with $ψ = ψ_{*} \neq = 0$ on a fixed neighbourhood of $t = 0$ , and a real observation test with $\int f d^{4} x = 1$ . For all sufficiently small widths,

Ψ_{ε} (t) - Ψ_{0} (t) = - ε ψ_{*} (1 - \int_{0}^{1} χ (s) d s)

at every observation time. The switch contribution does not depend on the width. Equation (eq:diagnostic) therefore gives

⟨ d_{1, ε} - d_{1, 0}, t ⟩ = - ε ψ_{*} (1 - \int_{0}^{1} χ (s) d s) .

For the stated ramp, the coefficient is nonzero. Pairing with $t$ is continuous on $D_{C}$ ; hence the representative difference cannot be $o (ε)$ .

Equivalently, on $K$ ,

c_{1, ε} - c_{1, 0} = ε ψ_{*} (1 - \int_{0}^{1} χ (s) d s) \partial_{t} f .

This gives the same conclusion directly.

Displayed formulas in this passage

unlabeled · line 1842
$Ψ_{ε} (t) - Ψ_{0} (t) = - ε ψ_{*} (1 - \int_{0}^{1} χ (s) d s)$
unlabeled · line 1850
$⟨ d_{1, ε} - d_{1, 0}, t ⟩ = - ε ψ_{*} (1 - \int_{0}^{1} χ (s) d s) .$
unlabeled · line 1861
$c_{1, ε} - c_{1, 0} = ε ψ_{*} (1 - \int_{0}^{1} χ (s) d s) \partial_{t} f .$

Proposition: The general rate is not o(ε)

Displayed formulas in this passage

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