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An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations |
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| Authors: | Dongho Chae Peter Constantin Jiahong Wu |
| Affiliation: | 1. Department of Mathematics, College of Natural Science, Chung-Ang University, Seoul, 156-756, Republic of Korea 2. Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA 3. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, 74078, USA 4. Department of Mathematics, Chung-Ang University, Seoul, 156-756, Republic of Korea |
| 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. |
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