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We give an example of a well posed, finite energy, 2D incompressible activescalar equation with the same scaling as the surface quasi-geostrophic equationand prove that it can produce finite time singularities. In spite of itssimplicity, this seems to be the first such example. Further, we constructexplicit solutions of the 2D Boussinesq equations whose gradients growexponentially in time for all time. In addition, we introduce a variant of the2D Boussinesq equations which is perhaps a more faithful companion of the 3Daxisymmetric Euler equations than the usual 2D Boussinesq equations.

An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations