Formation of Unstable Shocks for 2D Isentropic Compressible Euler

In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler azimuthal symmetry. More specifically, initial data that when viewed self-similar coordinates, converges asymptotically to $$C^{\frac{1}{5}}$$ solution Burgers’ equation. Moreover, show behavior is stable $$C^8$$ modulo a two dimensional linear subspace. Under symmetry assumption, one cannot impose additional assumptions order isolate corresponding manifold leading stability: rather, rely on modulation variable techniques conjunction with Newton scheme.