Self-similar solutions describe many singularities in nonlinear PDE. In this talk I will discuss two recent constructions of exactly self-similar blow-up for equations whose linear part is dispersive.

The first concerns the defocusing nonlinear Schrödinger equation in the energy supercritical regime. Bourgain conjectured in 2000 that solutions with smooth data are global. However, Merle, Raphaël, Rodnianski and Szeftel disproved this in dimensions 5, 6, 8 and 9 by building on imploding solutions of compressible Euler. Using a new mechanism, we construct smooth, compactly supported initial data that blow up in finite time in dimensions 3, 4 and 5.  

The second concerns patch solutions of the generalized SQG equations, a family of active scalar equations that interpolates between 2D Euler and the surface quasi-geostrophic equation. We construct a self-similar patch whose boundary is a smooth graph that develops a corner in finite time. The construction is perturbative in the slope of the corner and is driven by the dispersive character of the linearized equation. 

This is joint work with Kevin Buck, Alvaro Carballeira, Marcus Pasquariello and Jia Shi.