UCLA

We study the problem of finite time singularities in the isentropic compressible Euler equations in with radial symmetry. We focus on implosions, where a flow directed towards the origin experiences finite time blowup in density. Smooth and self-similar imploding solutions to the isentropic, compressible Euler equations with radial symmetry, inspired by the work of Guderley, have garnered recent mathematical interest. However, these smooth imploding solutions are shown to be numerically unstable and difficult to compute in practice. On the other hand, the imploding solution of Kidder has a closed form solution and is numerically computable. But, it is unbounded in the far field. We consider the Kidder implosion with a continuous constant density cutoff, referred to as the cutoff implosion solution, in this work.From studying the characteristics, we predict an expanding wave to emerge from the cutoff position. The expanding wave has its boundary described by the characteristics, which depend on the local speed of sound. The behavior of the cutoff implosion solution dramatically changes from one dimension to higher dimensions. In one dimension, a non-centered rarefaction emerges from the cutoff and suppresses the implosion. In higher dimensions, the expanding wave is not a classical rarefaction. However, the expanding wave still swallows up the inner Kidder solution and the implosion still occurs. This composite blowup has an asymptotic self-similar form that has an interpretation in terms of Guderley-type self-similarity structure. Consistent with prior rigorous results for these equations, the blowup does not involve mass concentration and exhibits a critical Lp space for the concentration of the density, depending on the dimension of space.The theoretical predictions made are supported by direct numerical simulations of the compressible Euler equations. The Lagrangian leapfrog numerical scheme is used to perform these computations. This scheme is chosen in particular to avoid numerical diffusion. In higher dimensions, adaptive mesh refinement is employed to resolve the implosion with high fidelity.