Self-similar imploding solutions compressible Navier-Stokes and higher dimensional NLS

We will talk about singularity formation for the 3D isentropic compressible Euler and Navier-Stokes and for higher dimensional supercritical defocusing NLS. Finite time singularities for radially symmetric data were found in the work of Merle, Raphaël, Rodnianski and Szeftel for both equations, based on self-similar imploding solutions. We will construct new self-similar profiles corresponding to larger self-similar exponents. In the case of 3D compressible Euler and Navier-Stokes, this also allows to treat different values of the adiabatic constant (in particular, gives the first such result for monatomic gases, where the adiabatic constant is 5/3). We will also explain how to upgrade the radially symmetric stability analysis to the nonradial case, which allows us to show singularity formation for nonradial initial data, and in the periodic setting. If time permits, we will also discuss other recent strategies that have been used to find self-similar profiles in fluids equations