This talk addresses the problem of singularity formation in solutions of the 3D compressible barotropic Navier-Stokes equation and of the energy supercritical defocusing nonlinear Schrödinger equation. I will explain the recent results of F. Merle, P. Raphaël, I. Rodnianski, and J. Szeftel that link this problem to the compressible Euler dynamics showing that in some range of parameters both models admit finite time blow up solutions governed by appropriate self-similar solutions of the underlying Euler equation. While for the compressible Navier-Stokes equation the existence of finite time blow up solutions was already known, for the nonlinear Schrödinger equation this is the first result of formation of singularities in the defocusing case.

[After Frank Merle, Pierre Raphaël, Igor Rodnianski and Jérémie Szeftel]