Bounded binary coagulation-fragmentation is globally well posed in weighted total variation

Since ${\mu }_t$ is positive, $\mathrm{\parallel }{\mu }_t{\mathrm{\parallel }}_w=M_0({\mu }_t)+M_2({\mu }_t)$. Applying [F3] for $0\le t\le T$ gives the displayed finite bound.

Suppose a maximal solution had a finite endpoint $T_{\ast }$. The model definition [F1] identifies its vector field, and the bound in step 1.1 places it in a fixed weighted-norm ball. By [F5], $\mathrm{\parallel }F({\mu }_t){\mathrm{\parallel }}_w\le C$ there, so $\mathrm{\parallel }{\mu }_t-{\mu }_s{\mathrm{\parallel }}_w\le C\mathrm{\mid }t-s\mathrm{\mid }$; hence ${\mu }_t$ is Cauchy as $t\uparrow T_{\ast }$. Completeness [F4] gives a limit ${\mu }_{\ast }$. Positivity is closed: if ${\mu }_n\ge 0$ and ${\mu }_n\to {\mu }_{\ast }$ in weighted total variation, then ${\mu }_{\ast }^{-}(E)\le \mathrm{\parallel }{\mu }_{\ast }-{\mu }_n{\mathrm{\parallel }}_{TV}\to 0$.

Restart the local construction [F2] at ${\mu }_{\ast }$. Its solution agrees on the overlap by local uniqueness and extends the original solution beyond $T_{\ast }$, a contradiction. Repeating this continuation excludes every finite right endpoint. If two global solutions have the same initial measure, local uniqueness in [F2] makes them agree successively on each compact time interval.