The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?

The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain $\varOmega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\boldsymbol{u}\cdot\hat{\boldsymbol{n}}|_{\partial\varOmega} = 0$ . It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B =∇ q×∇ θ, provided B has no null points initially: $\boldsymbol{\omega} = \operatorname{curl}\boldsymbol {u}$ is the vorticity and q= ω ⋅∇ θ 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 (Phys. Fluids 12:744–746, 2000 ) on the non-existence of Clebsch potentials in the neighbourhood of null points. [ABSTRACT FROM AUTHOR]