An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we constr...

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Bibliographic Details
Published in:Journal of mathematical fluid mechanics 2014-09, Vol.16 (3), p.473-480
Main Authors: Chae, Dongho, Constantin, Peter, Wu, Jiahong
Format: Article
Language:English
Subjects:
Classical and Continuum Physics
Fluid- and Aerodynamics
Mathematical Methods in Physics
Physics
Physics and Astronomy
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Summary:We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.
ISSN:1422-6928
1422-6952
DOI:10.1007/s00021-014-0166-5