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

Small Scales and Singularity Formation in Fluid Dynamics

“…There are a few more recent papers that have contributed towards understanding the hyperbolic point blow up scenario. Two-dimensional simplified models of the 2D Boussinesq system have been considered in [40] and in [53]. In both cases, the derivative forcing term in (2.2) is replaced by a simpler sign-definite approximation θ…”

“…x 1 , and the Biot-Savart law is replaced by a simpler version u = (−x 1 Ω(x, t), x 2 Ω(x, t)). In [40], Ω takes form similar to the 2D Euler example (3.6). In [53], Ω is closely related but is also chosen to keep u incompressible.…”

“…In [53], Ω is closely related but is also chosen to keep u incompressible. Both papers prove finite time blow up, [40] by a sort of barrier argument while the argument [53] deploys an appropriate Lyapunov-type functional.…”

Blow Up Of A Hyperbolic SQG Model

“…Independently and simultaneously, Hoang-Orcan-Radosz-Yang [11] proved blow up of the same equation (8)(9) with a different Biot-Savart law (13) u(x, t)…”