under the flow induced by $u$. To be more precise, define $A _{\underline{r},\overline{r}}$ to be the annular region $$\begin{aligned}
A _{\underline{r},\overline{r}}= \{ x\in\mathbb R^2 \colon \underline{r} < {\left|x\right|} < \overline{r} \} \,.\end{aligned}$$ Fixing $0<   r_0 < r_1$; then, if $\eta_u$ is the solution to ${\ensuremath{\partial}}_t \eta_u = u \circ \eta_u$ for $t>t_0$ with $\eta_u(x,t_0)=x$, define the time dependent domain $$\label{Omega}
\Omega(t)= \eta_u(A_{r_0,r_1},t)\,.$$ Now set $0< R_0 < r_0 < r_1 <R_0$ and let $K>0$. Assuming that ${\left|u\right|}\leq K$ for all $(x,t)\in A _{R_0,R_1}\times [t_0,T_*)$, then it follows that $$\Omega(t)\subset A_{R_0,R_1} \quad \text{ for } \quad t \in [t_0,T_*]\,,$$ so long as ${\left|T_*-t_0\right|}$ is assumed to be sufficiently small (depending or $r_0$, $r_1$, $R_0$, $R_1$ and $K$). Then given a solution $(a,b,P)$ to the system , we relate these to solutions to via the transformation , restricted to the domain $\Omega$ given in .

Brief historical overview
-------------------------

The formation of shocks is a classical problem in hyperbolic PDE. The first rigorous proof of shock formation is due to the pioneering work of Lax [@Lax1964] that employed invariants devised by Riemann [@Ri1860] and the method of characteristics. The work of Lax was further generalized and refined by John [@John1974], Liu [@Li1979], 