R$. Thus the initial data leading to the unstable shock profiles ${\overline}W_i $ for $i>1$ are described by an unstable manifolds of finite codimension.

Our main objective in this work is to identify an analogous unstable manifold, $\mathcal{M}_U$, for the compressible Euler equations which lead to unstable blowup dynamics according to the profile ${\overline}{W}_2$. Unlike the case for Burgers described above, the specification of $\mathcal{M}_U$ is not as explicit as that described above, and must be found via very careful Newton scheme.

Rough statement of main theorem
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In this paper, we prove the existence of asymptotically self similar solutions to 2D isentropic compressible Euler equations under azimuthal symmetry that under the appropriate self-similar transformations are described by the self-similar Burgers’ profile ${\overline}W_2$:

There exists initial data $(a_0,b_0, P_0)$ in $C^8$ for which the corresponding solutions $(a,b,P)$ to develop a $C^{\frac15}$-cusp singularity in finite time. At blow-up, the solutions $(a,b,P)$ form singularity at a unique angle; moreover, the singularities may be described in terms of the self-similar Burgers’ profile ${\overline}W_2$ in a manner made precise in Theorem \[thm:general\]. The behavior described is stable in $C^8$ with regards to the initial data modulo a two dimensional linear subspace.

We note that 