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

John D. Gibbon, Edriss S. Titi
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 : $\bom = curl \bu$ 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.
View original: http://arxiv.org/abs/1211.3811