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

The limiting Cauchy problems

Propagation and the fixed-support convergence theorem › The limiting Cauchy problems · lines 929–969

X-D-ST@1#eq:limit-cauchy, X-D-ST@1#eq:delta-data

Define $w_{0}, v_{0}$ , smooth on the right-hand closed slab, by

□ w_{0}^{ρ} □ v_{0}^{ρ} = - \partial_{μ} a^{μ ρ}, = - \partial_{μ} (A^{μ ρ} + a^{μν} \partial_{ν} w_{0}^{ρ}), w_{0}^{ρ} (0) v_{0}^{ρ} (0) = 0, = 0, \partial_{t} w_{0}^{ρ} (0) \partial_{t} v_{0}^{ρ} (0) = v_{1}^{ρ}, = v_{2}^{ρ} .

Their data are determined by Lemma lem:profile, not chosen independently.

The ordinary flat Cauchy problem constructs these solutions successively. For example, if $s_{j}^{ρ}$ denotes the corresponding right-hand source and $u_{1} = w_{0}, u_{2} = v_{0}$ , then

u_{j}^{ρ} (δ) \partial_{t} u_{j}^{ρ} (δ) = S (δ) v_{j}^{ρ} + \int_{0}^{δ} S (δ - s) s_{j}^{ρ} (s) d s, = C (δ) v_{j}^{ρ} + \int_{0}^{δ} C (δ - s) s_{j}^{ρ} (s) d s .

Thus no initial-layer datum is silently lost before propagation to the observation region.

Displayed formulas in this passage

X-D-ST@1#eq:limit-cauchy · line 933
$□ w_{0}^{ρ} □ v_{0}^{ρ} = - \partial_{μ} a^{μ ρ}, = - \partial_{μ} (A^{μ ρ} + a^{μν} \partial_{ν} w_{0}^{ρ}), w_{0}^{ρ} (0) v_{0}^{ρ} (0) = 0, = 0, \partial_{t} w_{0}^{ρ} (0) \partial_{t} v_{0}^{ρ} (0) = v_{1}^{ρ}, = v_{2}^{ρ} .$
X-D-ST@1#eq:delta-data · line 957
$u_{j}^{ρ} (δ) \partial_{t} u_{j}^{ρ} (δ) = S (δ) v_{j}^{ρ} + \int_{0}^{δ} S (δ - s) s_{j}^{ρ} (s) d s, = C (δ) v_{j}^{ρ} + \int_{0}^{δ} C (δ - s) s_{j}^{ρ} (s) d s .$

The limiting Cauchy problems

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