Manuscript Hawk02-A@1 · Hawk02-A.tex · proposition · 7fbae745

Proposition: Mass-ledger stability

Stage 2: the backreaction interface › A conditional ledger allowance · lines 579–605 · proof included or linked

Hawk02-A@1#prop:massstability, Hawk02-A@1#eq:massstability

Let $P$ be Lipschitz with constant $L$ on a compact mass interval $I \subset (0, \infty)$ . Suppose $M$ and $M$ remain in $I$ for $0 \leq u \leq T$ and satisfy

M ˙ = - P (M) + r (u), M ˙ = - P (M), ∣ r (u) ∣ \leq b (u),

where $b$ is integrable. Then

∣ M (u) - M (u) ∣ \leq e^{Lu} ∣ M (0) - M (0) ∣ + \int_{0}^{u} e^{L (u - s)} b (s) d s .

Proof. Subtract the equations. The upper derivative of the absolute difference is bounded by $L ∣ M - M ∣ + b$ . Gronwall's inequality gives (eq:massstability).

Displayed formulas in this passage

Hawk02-A@1#eq:massstability · line 590
$∣ M (u) - M (u) ∣ \leq e^{Lu} ∣ M (0) - M (0) ∣ + \int_{0}^{u} e^{L (u - s)} b (s) d s .$
unlabeled · line 584
$M ˙ = - P (M) + r (u), M ˙ = - P (M), ∣ r (u) ∣ \leq b (u),$

Nearby in this section

section: A conditional ledger allowance