APDE seminar@BCAM: Finite-time singularity via pendula for the forced 2D Boussinesq equation

Abstract

The blow-up takes place in a well-posedness regime, i.e., the force is uniformly bounded in time in some space where there is local existence of solutions. The singularity follows from a "vorticity layer cascade" mechanism based on an accumulated hysteresis  effect on the amplitude of the vorticity layers caused by the deformation of the density layers. The displacement and deformation of  those layers is governed by an infinite number of copies of the ODE that describes the movement of a pendulum. Furthermore, the density vanishes uniformly in space at a sequence of times that accumulate at  the blow-up point, giving the singularity a flickering nature. Besides, the vorticity, the density, the velocity, the force of the  vorticity equation and the force of the density equation of our solution will be compactly supported in space and their support will be uniform in time. In this talk, we will strive to describe the construction in detail and provide some ideas of the proof. Moreover, if time allows it, we will also comment how this construction may be used to prove a blow-up for the 2D non-homogeneous incompressible Euler equations.