Cumulative Euler flows

TL;DR

This paper investigates the potential for singularities, such as mass accumulation, in solutions to the compressible Euler equations for ideal gases, focusing on radial affine motions and their physical plausibility.

Contribution

It demonstrates that all affine cumulative solutions exhibit unphysical behavior and explores the mechanisms and conditions for physically acceptable mass accumulation in Euler flows.

Findings

Affine cumulative solutions are unphysical due to unbounded velocities.

Two mechanisms for mass accumulation are identified: inertial effects and adverse pressure gradients.

The paper discusses modifications to obtain physically acceptable accumulation behaviors.

Abstract

We consider the compressible Euler system for ideal gas flow in the absence of any forces except the internal thermodynamic pressure. In this setting, and in dimensions higher 1, it is known that wave-focusing can drive Euler solutions to amplitude blowup in finite time from bounded initial data. In the known cases (self-similar, radial flows \cites{gud,hun_60,jt3,laz,mrrs1,jls}) the primary flow variables are standard functions at time of blowup. It is natural to ask if the Euler system admits even more singular behavior, and specifically whether accumulation of mass, i.e., the appearance of a Dirac delta in the density field, is possible. We consider the class of radial affine motions \cites{mcvittie,sed, kell,sid_2014} which are conveniently obtained via a Lagrangian formulation. This class does include examples of cumulative behavior, and we observe that there are two distinct mechanisms for accumulation, due to inertial effects or adverse pressure gradients, respectively. However, we show that all affine cumulative solutions necessarily exhibit unphysical behavior due to initially unbounded velocity and/or acceleration in the far-field. We also analyze the behavior of characteristics in cumulative flows and consider concrete examples, including a class of 1-dimensional, non-affine flows. Finally, we discuss the possibility of modifying the known examples to obtain physically acceptable gas flows displaying accumulation.