Complex Dynamics of Wave-Character Transitions in Radially Symmetric Isentropic Euler Flows: Theory and Numerics

TL;DR

This paper analyzes the wave-character transitions in radially symmetric isentropic Euler flows, establishing theoretical restrictions and invariant domains, and supports findings with numerical simulations to understand singularity formation.

Contribution

It provides new insights into wave transitions in subsonic and inward regimes, refining invariance properties and identifying mechanisms absent in purely supersonic flows.

Findings

Refined invariance properties in outward supersonic regime

Discovery of asymmetric transition mechanisms in subsonic and inward regimes

Numerical simulations confirming theoretical wave dynamics and singularity conditions

Abstract

We investigate the qualitative dynamics of smooth solutions to the radially symmetric isentropic compressible Euler equations, focusing specifically on the evolution of rarefactive and compressive wave characters across three distinct configurations: the outward supersonic, subsonic, and inward supersonic regimes. For each case, we establish structural restrictions on wave-character transitions and identify invariant sign domains for gradient variables under specific initial data conditions. While our findings refine existing invariance properties in the outward supersonic regime, they reveal novel asymmetric transition mechanisms in the subsonic and inward regimes that are absent in purely supersonic expanding cases. Consequently, we derive sufficient conditions for finite-time singularity formation. To complement the analytical results where closed-form solutions are unavailable, we provide numerical experiments using a Semi-Discrete Lagrangian-Eulerian (SDLE) formulation. These simulations reproduce the predicted wave-character dynamics and offer qualitative evidence that supports our theoretical findings, providing a unified description of wave transitions in radially symmetric isentropic gas dynamics.