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

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.

9 pages; simplified a solution formula in section 4 and added a sentence on the time growth rate in the solution

References in corpus (2)

• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• On the blow-up problem for the axisymmetric 3D Euler equations

Cited by in corpus (9)

• Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation
• Finite-time Singularity Formation for Strong Solutions to the Boussinesq System
• Finite time blow up in the hyperbolic Boussinesq system
• On the regularity of solutions to the 2D Boussinesq equations satisfying Type I conditions
• Blowup with vorticity control for a 2D model of the Boussinesq equations
• Global smooth solution to the 2D Boussinesq equations with fractional dissipation
• The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain
• Removing Type II singularities off the axis for the 3D axisymmetric Euler equations
• Analysis of a Singular Boussinesq Model