Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five

Abstract

We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u-|u{|}^{p-1}u=0$ in ${\mathbb{R}}^d$, for $(d,p)\in \{(3,27),(4,9),(5,7)\}$. The solutions are asymptotically self-similar, their $L^{\infty }$ norm grows at the scaling rate $(T-t{)}^{-1/(p-1)}$, and their critical Sobolev norm ${\dot{H}}^{s_c}$ diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.

44 pages. Python code for the computer-assisted argument is included as suplementary files