Proposition: Propagation of the exit-data limit

For every spatial Sobolev order $r$,

after the fixed spatial localization. All spacetime derivatives converge at rate $O(\epsilon )$ on compact subsets of $[\delta ,T]\times {\mathbb{R}}^3$. In particular, the ramp data at $t=\delta$ converge to (eq:delta-data).

Proof. For $t\ge \epsilon$, ${\sigma }_{\epsilon }=1$ and every positive-order ramp derivative vanishes. Put $w_j={\mathsf{X}}_{j,\epsilon }-{\mathsf{X}}_{j,0}$. The layer lemma and smoothness of the right-hand solutions give $O(\epsilon )$ Cauchy data for both differences at $t=\epsilon$ in every spatial Sobolev norm.

The first difference satisfies $\square w_1=0$. The homogeneous wave formula bounds $w_1,{\partial }_t{w}_1$ by those data uniformly on the fixed interval. For the second difference,

The source is $O(\epsilon )$ in every spatial Sobolev norm: use the first estimate and ${{\partial }_t}^2{w}_1=\Delta w_1$. The inhomogeneous Duhamel formula gives the second estimate.

Higher time derivatives follow inductively from these equations, using higher spatial norms and derivatives of the fixed smooth coefficients. Sobolev embedding gives the asserted smooth convergence. The constants are independent of the initial time $\epsilon \in (0,\delta )$.