Amplitude blowup in compressible Euler flows without shock formation

TL;DR

This paper demonstrates that in 3D compressible Euler flows, pressure blowup can occur without shock formation, revealing complex behaviors in flow collapse scenarios and challenging traditional shock formation understanding.

Contribution

It proves the existence of continuous 3D flows that blow up at collapse but remain continuous, even with inward motion, highlighting subtlety in shock formation mechanisms.

Findings

Pressure can blow up without shock formation in 3D Euler flows.

Continuous flows can experience collapse while remaining smooth.

Flow behavior near collapse is more complex than previously understood.

Abstract

Recent works have demonstrated that continuous self-similar radial Euler flows can drive primary (non-differentiated) flow variables to infinity at the center of motion. Among the variables that blow up at collapse is the pressure, and it is unsurprising that this type of behavior can generate an outgoing shock wave. In this work we prove that there is an alternative scenario in which an incoming, continuous 3-d flow suffers blowup, including in pressure, and yet remains continuous beyond collapse. We verify that this behavior is possible even in cases where the fluid is everywhere moving toward the center of motion at time of collapse. The results underscore the subtlety of shock formation in multi-dimensional flow.