Abstract
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.
Original language | English (US) |
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Pages (from-to) | 473-480 |
Number of pages | 8 |
Journal | Journal of Mathematical Fluid Mechanics |
Volume | 16 |
Issue number | 3 |
DOIs | |
State | Published - Sep 2014 |
All Science Journal Classification (ASJC) codes
- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics
Keywords
- 2D Boussinesq equations
- Explicit solutions
- Inviscid model
- Singularity