An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

Abstract

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.

Blowup with vorticity control for a 2D model of the Boussinesq equations

“…On the other hand, the system (1) exhibits some mathematical features of great interest. It is well-known that the three-dimensional axisymmetric Euler equations in a cylinder are almost identical to (1) at least at points away from the cylinder axis [14]. That is, (1) contains an analogue of the 3D Euler vorticity stretching mechanism in the right-hand side term ρ x 1 of the first equation of (1).…”

“…The two-dimensional Boussinesq equations for the vorticity ω and density ρ ω t + u · ∇ω = ρ x 1 ρ t + u · ∇ρ = 0 (1) with velocity field u = (u 1 , u 2 ) given by (2) u = ∇ ⊥ (−∆) −1 ω are an important system of partial differential equations arising in atmospheric sciences and geophysics, where the system models an incompressible fluid of varying, temperature dependent density subject to gravity. On the other hand, the system (1) exhibits some mathematical features of great interest.…”

“…It is well-known that the three-dimensional axisymmetric Euler equations in a cylinder are almost identical to (1) at least at points away from the cylinder axis [14]. That is, (1) contains an analogue of the 3D Euler vorticity stretching mechanism in the right-hand side term ρ x 1 of the first equation of (1). The question whether solutions of the 3D Euler equations blow up in finite time from smooth data with finite energy is a profound, as of yet unanswered question [4].…”

Removing Type II Singularities Off the Axis for the Three Dimensional Axisymmetric Euler Equations

“…We say the vorticity ø has Type I blow-up at t * in a domain D ⊂ R 3 if (1. 7) lim sup t→t * (t * − t) ø(t) ∞(D) < +∞.…”

Finite time blow up in the hyperbolic Boussinesq system

“…This set up corresponds to Hou-Luo scenario turned by π/2 : x 1 corresponds to z and x 2 to r, and for the right initial data we expect very fast growth of ω at the origin. We note that, naturally, the problem of global regularity vs finite time blow up for the system (1), (2), (3) is also open and well known. It is appears, for example, as one of the "eleven great problems of mathematical hydrodynamics" in [15].…”