analogous results exist for the Burger’s equation with traversal viscosity [@CoGhMa2018], the Prandtl equations [@CoGhIbMa18; @CoGhMa19] and the Burgers-Hilbert equation [@yang2020shock]. We also note that the formation of *unstable* shocks (defined and discussed below) in the context of Bourgain-Wang solutions to NLS was obtained in [@MR3086066] through virial type identities and backwards integration techniques. These papers however rely on a symmetry to constrain the position of the singularity which leads to a comparatively simple classification of initial data leading to unstable blow up profiles. Isentropic Euler does not satisfy analogous symmetries leading us to develop a new shooting method in order to describe initial data leading to unstable blowup. We believe that techniques developed are suitably malleable and could find potential use in proving the existence of unstable blowup for other PDE.

Statement of main theorem
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Riemann invariants
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Before we can state our main theorem, we must first introduce the concept of Riemann invariants, since it is our aim to show that we can prescribe initial data such that one of the Riemann invariants shocks according to the self-similar profile ${\overline}W_1$.

As was done in [@BuShVi2019], in order to diagonalize the system - and isolate the Burgers-like behavior 