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 C15 self-similar solution to the Burgers’ equation. Moreover, we show the behavior is stable in C8 modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding manifold of initial data leading to stability: rather, we rely on modulation variable techniques in conjunction with a Newton scheme.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics