A Solution to the Focusing 3d Nls That Blows up on a Contracting Sphere

We rigorously construct radial H solutions to the 3d cubic focusing NLS equation i∂tψ + ∆ψ + 2|ψ|ψ = 0 that blow-up along a contracting sphere. With blow-up time set to t = 0, the solutions concentrate on a sphere at radius ∼ t but focus towards this sphere at the faster rate ∼ t. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov [2] in 1975 and independently later in Holmer-Roudenko [5] in 2007, where it was demonstrated to be consistent with all conservation laws of this equation. In the latter paper, it was proposed as a solution that would yield divergence of the Lx norm within the “wide” radius ∼ ‖∇u(t)‖ Lx but not within the “tight” radius ∼ ‖∇u(t)‖ −2 Lx , the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail in Sulem-Sulem [13, Chapter 7].