Instability of Self-Similar Implosions for the Compressible Euler Equations

We prove that all known regular self-similar implosion solutions of the isentropic compressible Euler equations are unstable to shock formation. The perturbations can be chosen to be small in a class determined by the regularity threshold of the background. The proof relies on an amplification mechanism which occurs along the surface of the backward sonic cone from the implosion point. Motivated by our main results, we formulate an implosion censorship conjecture for the compressible Euler equations, stating that classical implosions are unstable to shock formation.