The 3D incompressible Euler equations with a passive scalar: a road to blow-up?

TL;DR

This paper investigates conditions under which smooth solutions to the 3D incompressible Euler equations with a passive scalar develop singularities, highlighting the role of a specific vector field and potential vorticity in predicting blow-up.

Contribution

It introduces a new blow-up criterion involving the development of null points in a vector related to the scalar and vorticity, emphasizing the passive scalar's role in singularity formation.

Findings

Smooth solutions blow up if a null point develops in the vector B.

The passive scalar is crucial in detecting singularity formation.

The criterion relates to the behavior of potential vorticity and null points.

Abstract

The 3D incompressible Euler equations with a passive scalar $\theta$ are considered in a smooth domain $\Omega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\bu\cdot\bhn|_{\partial\Omega} = 0$. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector $\bB = \nabla q\times\nabla\theta$, provided $\bB$ has no null points initially\,: $\bom = \mbox{curl}\,\bu$ is the vorticity and $q = \bom\cdot\nabla\theta$ is a potential vorticity. The presence of the passive scalar concentration $\theta$ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.