Chae, Dongho; Constantin, Peter; Wu, Jiahong An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations. (English) Zbl 1307.35213 J. Math. Fluid Mech. 16, No. 3, 473-480 (2014); erratum ibid. 16, No. 3, 481 (2014). 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. Cited in 1 ReviewCited in 9 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids Keywords:inviscid model; singularity; explicit solutions; 2D Boussinesq equations PDF BibTeX XML Cite \textit{D. Chae} et al., J. Math. Fluid Mech. 16, No. 3, 473--480 (2014; Zbl 1307.35213) Full Text: DOI arXiv References: [1] Blumen, W., Uniform potential vorticity flow, part I. theory of wave interactions and two-dimensional turbulence, J. Atmos. Sci., 35, 774-783, (1978) [2] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F., Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal., 208, 805-909, (2013) · Zbl 1293.35234 [3] Chae, D., On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity, 21, 2053-2060, (2008) · Zbl 1186.35155 [4] Constantin, P.; Majda, A.; Tabak, E., Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity, 7, 1495-1533, (1994) · Zbl 0809.35057 [5] Denisov, S.: Double-exponential growth of the vorticity gradient for the two-dimensional Euler equation. Proc. Amer. Math. Soc. (to appear) · Zbl 1315.35150 [6] Gill A.E.: Atmosphere-Ocean Dynamics. Academic Press, New York (1982) [7] Held, I.; Pierrehumbert, R.; Garner, S.; Swanson, K., Surface quasi-geostrophic dynamics, J. Fluid Mech., 282, 1-20, (1995) · Zbl 0832.76012 [8] Kiselev, A., Sverak, V.: Small scale creation for solutions of the incompressible two dimensional Euler equations. arXiv:1310.4799v2[math.AP], 24 Oct 2013 [9] Luo, G., Hou,T.: Potentially singular solutions of the 3D incompressible Euler equations. arXiv:1310.0497v1[physics. flu-dyn.], 1 Oct 2013 · Zbl 0809.35057 [10] Majda A.J., Bertozzi A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, UK (2001) · Zbl 0983.76001 [11] Pedlosky J.: Geophysical Fluid Dynamics. Springer, New York (1987) · Zbl 0713.76005 [12] Zlatos, A.: Exponential growth of the vorticity gradient for the Euler equation on the torus. arXiv:1310.6128v1[math.AP], 23 Oct 2013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.