Abstract
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.
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Acknowledgements
The authors would like to thank a referee for some very helpful comments. John Gibbon also thanks the Isaac Newton Institute for Mathematical Sciences Cambridge for its hospitality (July–December 2012) on the programme ‘Topological Dynamics in the Physical and Biological Sciences’, under whose auspices this work was done. He would also like to thank Darryl Holm of Imperial College London for several discussions on this problem. The work of E.S. Titi is supported in part by the Minerva Stiftung/Foundation and also by the NSF grants DMS-1009950, DMS-1109640 and DMS-1109645.
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Communicated by Peter Constantin.
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Gibbon, J.D., Titi, E.S. The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?. J Nonlinear Sci 23, 993–1000 (2013). https://doi.org/10.1007/s00332-013-9175-4
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DOI: https://doi.org/10.1007/s00332-013-9175-4