Formation of Singularity for Compressible Euler Equations Outside a Ball in 3-D

Abstract

The initial boundary value problem for a compressible Euler system outside a ball in $\mathbf{R}^3$ is considered in this paper. Assuming the initial data have small and compact supported perturbations near a constant state, we show that the solution will blow up in a finite time, and the lifespan estimate can be estimated by the small parameter of the initial perturbations. To this end, a “tricky” test function admitting good behavior is introduced.