---
abstract: 'In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler in azimuthal symmetry. More specifically, we construct initial data that when viewed in self-similar coordinates, converges asymptotically to the unstable $C^{\frac15}$ self-similar solution to the Burgers’ equation. Moreover, we show the behavior is stable in $C^8$ modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding unstable manifold: rather, we rely on modulation variable techniques in conjunction with a Newton scheme.'
author:
- '[ [^1] ]{}'
- '[[^2] ]{}'
bibliography:
- 'euler.bib'
title: '[**Formation of unstable shocks for 2D isentropic compressible Euler** ]{}'
---

Introduction
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Setup of Compressible Euler under azimuthal symmetry
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In this paper we study asymptotically self-similar formation of unstable shocks for the 2D isentropic compressible Euler equations under azimuthal symmetry. The 2D isentropic compressible Euler equations take the form

\[eq:Euler\] $$\begin{aligned}
\partial_t (\rho u) + {\ensuremath{\mathrm{div\,}}}(\rho\, u \otimes u) + \nabla p(\rho) &= 0 \,,  \label{eq:momentum} \\
\partial_t \rho  + {\ensuremath{\mathrm{div\,}}}(\rho u)&=0 \,,  \label{eq:mass}\end{aligned}$$

where $u :\mathbb{R}^2  \times \mathbb{R}  \to \mathbb{R}^2  $ is the velocity of the fluid, $\rho: \mathbb{R}^2  \times \mathbb{R}  \to \mathbb{R}  _+$ 