is the density, and $p: \mathbb{R}^2  \times \mathbb{R}  \to \mathbb{R}  _+$ is the pressure defined by the ideal gas law $$p(\rho) := \tfrac{1}{\gamma} \rho^\gamma\,, \qquad \gamma >1 \,.$$ The associated sound speed $\sigma$ is given by $\sigma=\rho^\lambda$ where $\lambda=\frac{\gamma-1}{2}$.

It was shown in [@BuShVi2019], that if one imposes the following azimuthal symmetry $$\label{e:symmetry}
u(x,t)\cdot \frac{x}{{\left|x\right|}}= r a(\theta,t), \quad u(x,t)\cdot \frac{x^{\perp}}{{\left|x\right|}}=r b(\theta,t),\quad \rho=r^{\frac2{\gamma-1}}  P(\theta,t),$$ where $(r,\theta)$ are the usual polar coordinates, then the equations reduce to the 1D system of equations

\[eq:Euler:polar3\] $$\begin{aligned}
 \label{g3_a_evo}
\left(\partial_t  + b\partial_{\theta}\right) a  + a^2-b^2+  \lambda^{-1} P^{2 \lambda }   &=0 \\ \label{g3_b_evo}
\left(\partial_t  + b\partial_{\theta}\right)b+2a b+  P^{ 2 \lambda -1}\partial_\theta P&=0 \\ \label{g3_P_evo}
\left(\partial_t  + b\partial_{\theta}\right) P+  \tfrac{\gamma}{ \lambda } a P+  P  {\ensuremath{\partial}}_\theta b &=0 \, .\end{aligned}$$

An important difference between Euler under azimuthal rather than radial symmetry is that azimuthal symmetry allows for the presence of non-trivial vorticity. We remark that it was shown in [@BuShVi2019], that the system is locally well-posed in $C^n$ for any $n\geq 1$.

In order to avoid issues regarding the irregularity at the origin $r=0$, and in order to ensure finite kinetic-energy, following [@BuShVi2019], we can exploit locality and restrict the solution to the push forward of an annulus 