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

Abstract:

We rigorously construct radial $H^1$ solutions to the 3d cubic focusing NLS equation $i\partial _t \psi + \Delta \psi + 2 |\psi |^2\psi =0$ that blow-up along a contracting sphere. With blow-up time set to $t=0$, the solutions concentrate on a sphere at radius $\sim t^{1/3}$ but focus towards this sphere at the faster rate $\sim t^{2/3}$. 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 $L_x^3$ norm within the “wide” radius $\sim \|\nabla u(t)\|_{L_x^2}^{-1/2}$ but not within the “tight” radius $\sim \|\nabla u(t)\|_{L_x^2}^{-2}$, the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail by Sulem-Sulem in 1999.