From Instability to Singularity Formation in Incompressible Fluids
In this talk, I will describe a new mechanism for singularity formation in the 2d Boussinesq system and in the 3d incompressible Euler equations. In the Boussinesq case, the singularity mechanism arises as a second order effect on the classical Rayleigh–Bénard instability, and the initial data we choose is smooth except at one point, where it has Hölder continuous first derivatives. In addition, the solution is smooth in the angular variable at the blow-up point, which was a fundamental obstruction in previous works. I will finally describe how these considerations translate to a singularity formation scenario for the 3d incompressible Euler equations, based on the Taylor–Couette instability. This is joint work with Tarek Elgindi (Duke University).