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


  • The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain with no-normal-flow boundary conditions = 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: = {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. © 2013 Springer Science+Business Media New York.

author list (cited authors)

  • Gibbon, J. D., & Titi, E. S.

citation count

  • 7

publication date

  • June 2013