Manuscript VI@1 · VI.tex · section · b5b60afb

The finite-time inverse

Finite-time harmonic coordinates and second-order descent › The finite-time inverse · lines 908–929

VI@1#eq:Gzero

For a smooth source $b$ , let $G_{□, 0} b$ denote the solution of

□_{η} u = b, u (0) = \partial_{t} u (0) = 0.

On $0 \leq t \leq T$ ,

(G_{□, 0} b) (t) = - \int_{0}^{t} \frac{sin (( t - s ) ∣ D ∣ )}{∣ D ∣} b (s) d s .

For sources without spatial decay, the formula is interpreted locally using finite propagation and compatible compact cutoffs. Values on a compact target set depend only on a compact truncated past region. Ordinary wave Cauchy theory justifies this construction BGP.

No inverse from an infinite past is needed.

Displayed formulas in this passage

VI@1#eq:Gzero · line 915
$(G_{□, 0} b) (t) = - \int_{0}^{t} \frac{sin (( t - s ) ∣ D ∣ )}{∣ D ∣} b (s) d s .$
unlabeled · line 911
$□_{η} u = b, u (0) = \partial_{t} u (0) = 0.$