A solution to the focusing 3d NLS that blows up on a contracting sphere

Abstract

We rigorously construct radial H 1 H^1 solutions to the 3d cubic focusing NLS equation i ∂ t ψ + Δ ψ + 2 | ψ | 2 ψ = 0 i\partial _t \psi + \Delta \psi + 2 |\psi |^2\psi =0 that blow-up along a contracting sphere. With blow-up time set to t = 0 t=0 , the solutions concentrate on a sphere at radius ∼ t 1 / 3 \sim t^{1/3} but focus towards this sphere at the faster rate ∼ t 2 / 3 \sim t^{2/3} . Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov in 1975 and independently later by Holmer-Roudenko 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 L x 3 L_x^3 norm within the “wide” radius ∼ ‖ ∇ u ( t ) ‖ L x 2 − 1 / 2 \sim \|\nabla u(t)\|_{L_x^2}^{-1/2} but not within the “tight” radius ∼ ‖ ∇ u ( t ) ‖ L x 2 − 2 \sim \|\nabla u(t)\|_{L_x^2}^{-2} , the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail by Sulem-Sulem in 1999.