3-d Shock-Free Compressible Euler Flows with Amplitude Blowup and Pile-Up of Characteristics

Abstract

Recent works have demonstrated that self-similar radial Euler flows of an ideal polytropic gas in 2 and 3 dimensions can drive primary (non-differentiated) flow variables to infinity at the center of motion. Unsurprisingly, this type of behavior can generate an outgoing shock wave from the point of collapse. In this work we prove that there is an alternative scenario possible in 3-d flow: an incoming, radial \(C^1\) flow suffers blowup at a single point, including in density and pressure, and yet remains \(C^1\) beyond collapse. No shock is generated despite the fact that the density gradient \(\nabla _{{\textbf{x}}}\rho \), as well as the density of characteristics, becomes unbounded at the center of motion at time of blowup. Furthermore, 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, in which case also \({{\,\textrm{div}\,}}_{{\textbf{x}}}{\textbf{u}}\rightarrow -\infty \). These phenomena underscore the subtlety of shock formation in multi-d flow.