Theorem: Transmission characterization

The map $X_{\mathrm{sh}}$ is the unique formal piecewise smooth continuous map equal to $x$ for $t<0$ and satisfying

as a distribution, where the flux is formed first as a locally integrable piecewise smooth function. Its completion-normal flux is continuous across $\Sigma$.

Proof. The derivative of a continuous piecewise smooth coordinate function is a piecewise smooth locally integrable function; there is no coordinate-jump delta in that first derivative. Thus the displayed flux is defined coefficientwise by ordinary multiplication of piecewise smooth functions.

Its divergence has a surface delta whose coefficient is the jump of its normal component. On the negative side that component is ${\eta }^{0\rho }={{\delta }^{\rho }}_0$. On the positive side its initial value is

Tangential positive-order derivatives vanish initially, because the positive-order positions are identically zero on $\Sigma$. The jump vanishes exactly for (eq:velocities). The bulk equations are (eq:limit-cauchy), proving (eq:transmission).

Conversely, continuity forces the zero positive-order initial positions. Flux matching determines the two velocities successively. The smooth right-hand Cauchy problems then have unique solutions.