Blow up for the energy super critical defocusing NLS
The defocusing Non Linear Schrödinger equation iut=Δu−u|u|p−1 is a classical model of mathematical physics. For energy subcritical non linearities, Ginibre and Velo proved in the ’80s that all solutions are global in time and asymptotically behave like linear waves, and this was extended to the energy critical range in Bourgain’s breakthrough work in 1994. But the proof cannot address the energy super critical case for which global existence vs singularity formation is a classical problem in the field. I will explain in this lecture the construction of new unexpected highly oscillatory blow up mechanisms in a suitable energy super critical range of parameters as recently obtained in collaboration with Merle (IHES), Rodnianski (Princeton), and Szeftel (Sorbonne). The construction is deeply connected to the first description of viscous three dimensional compressible shock waves in fluid mechanics.