Finite time implosions of the multidimensional compressible Euler equations with a finite far-field cutoff

Abstract

We study the problem of finite time singularities in the isentropic compressible Euler equations in multiple dimensions with radial symmetry. Our initial data consists of the radially symmetric Kidder self-similar implosion with a continuous constant density cutoff. Motivated by our recent work on the one-dimensional problem, we present high-resolution radially symmetric adaptive mesh refinement simulations that when combined with asymptotic similarity solution analysis present a full picture of the ensuing dynamics of the implosion. From the cutoff point in the initial data, an expanding wave emerges. The endpoints of the wave follow the "imploding" and "exploding" characteristic speeds associated with the system of hyperbolic conservation laws. This wave has much more complex structure than the exact non-centered rarefaction wave that emerges in the one-dimensional problem. As in the one-dimensional case, the inner Kidder solution is swallowed up by the expanding wave as time approaches the singularity time. However, unlike the one-dimensional case, a blowup occurs along the point where the `imploding' side of the expanding wave attaches to the Kidder solution. This composite blowup has an asymptotic self-similar form that has an interpretation in terms of the Gurderley-type self-similarity structure. The numerical method for the PDE necessitates a Lagrangian scheme to avoid the numerical diffusion inherent in Eulerian methods. We use a leapfrog method coupled to recursive refinement of the mesh near the blowup, which allows for very high resolution simulations, resolving many orders of magnitude of the self-similar blowup. Consistent with prior rigorous results for these equations, the blowup does not involve mass concentration and exhibits a critical $L^p$ space for the concentration of the density, depending on the dimension of space.

arXiv comment23 pages, 17 figures