Lifespan Bounds and Super-Classical Propagation for Weak Solutions of the Three-Dimensional Compressible Euler Equations

Abstract

We establish an upper bound on the lifespan of bounded weak solutions to the three-dimensional isentropic compressible Euler equations on $[0,T)\times\mathbb R^3$, under a localized positivity condition on the initial data and the assumption that the essential support of the disturbance propagates into an undisturbed background state no faster than predicted by classical local well-posedness. We further show that any entropy-admissible bounded weak solution with bounded inverse density that persists beyond this upper bound must propagate into the undisturbed region at a strictly super-classical speed and that this accelerated propagation is necessarily accompanied by a jump discontinuity in an associated one-sided $L^\infty$-profile of the solution.