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

Abstract

The 3D incompressible Euler equations with a passive scalar $θ$ are considered in a smooth domain $Ω\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\bu\cdot\bhn|_{\partialΩ} = 0$. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector $\bB = \nabla q\times\nablaθ$, provided $\bB$ has no null points initially\,: $\bom = \mbox{curl}\,\bu$ is the vorticity and $q = \bom\cdot\nablaθ$ is a potential vorticity. The presence of the passive scalar concentration $θ$ 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.