# Mathematical analysis of pulsatile flow, vortex breakdown and instantaneous blow-up for the axisymmetric Euler equations

## TL;DR

This paper analyzes the stability and blow-up phenomena in axisymmetric Euler flows, showing rapid inflow can destabilize laminar profiles and non-zero axis vorticity prevents steady states, indicating potential singularities.

## Contribution

It provides a mathematical framework demonstrating conditions leading to instability and blow-up in axisymmetric Euler equations, using Frenet-Serret formulas and moving frames.

## Key findings

- Rapid inflow destabilizes laminar flow profiles.
- Non-zero vorticity on the axis prevents steady solutions.
- Conditions for instantaneous blow-up are identified.

## Abstract

The dynamics along the particle trajectories for the 3D axisymmetric Euler equations are considered. It is shown that if the inflow is rapidly increasing (pushy) in time, the corresponding laminar profile of the incompressible Euler flow is not (in some sense) stable provided that the swirling component is not zero. It is also shown that if the vorticity on the axis is not zero (with some extra assumptions), then there is no steady flow. We can rephrase these instability to an instantaneous blow-up. In the proof, Frenet-Serret formulas and orthonormal moving frame are essentially used.