Manuscript Em03-D@1 · Em03-D-v1.tex · section · 450d878a

Compact Cauchy truncation and time-slice test

Bridge A: compact bulk tests approaching the packet › Compact Cauchy truncation and time-slice test · lines 781–810

Em03-D@1#eq:compactpackettest

Choose a real $χ \in C_{c}^{\infty} ((- 2, 2))$ , equal to one on $[- 1, 1]$ . For $R \geq 1$ , put

q_{R} = χ (x / R) q, p_{R} = χ (x / R) p .

Let $s_{R}$ solve

D s_{R} = 0, s_{R} (0) = q_{R}, \partial_{t} s_{R} (0) = p_{R} .

Choose a smooth time function $η_{T}$ , equal to zero for $t \leq - T$ and one for $t \geq T$ , with $T > 0$ . Define

f_{R, T} = - \partial_{t}^{2} D (η_{T} s_{R}), F_{R, T} = F [f_{R, T}] .

Because

D (η_{T} s_{R}) = η_{T}^{''} s_{R} + 2 η_{T}^{'} \partial_{t} s_{R},

the reduced test is smooth and compact, with

∣ t ∣ \leq T, ∣ x ∣ \leq 2 R + T .

The full support includes the angular sphere $S^{2}$ .

Displayed formulas in this passage

Em03-D@1#eq:compactpackettest · line 795
$f_{R, T} = - \partial_{t}^{2} D (η_{T} s_{R}), F_{R, T} = F [f_{R, T}] .$
unlabeled · line 785
$q_{R} = χ (x / R) q, p_{R} = χ (x / R) p .$
unlabeled · line 789
$D s_{R} = 0, s_{R} (0) = q_{R}, \partial_{t} s_{R} (0) = p_{R} .$
unlabeled · line 802
$D (η_{T} s_{R}) = η_{T}^{''} s_{R} + 2 η_{T}^{'} \partial_{t} s_{R},$
unlabeled · line 806
$∣ t ∣ \leq T, ∣ x ∣ \leq 2 R + T .$