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

TL;DR

This paper constructs rigorous examples of solutions to the 3d focusing cubic NLS that blow up along a contracting sphere, confirming long-standing heuristic predictions about their dynamics and concentration rates.

Contribution

It provides the first rigorous construction of solutions with spherical blow-up behavior and specific concentration rates, validating previous heuristic and numerical studies.

Findings

Solutions blow up along a contracting sphere at radius ~ t^{1/3}

Concentration towards the sphere occurs at rate ~ t^{2/3}

Results confirm heuristic predictions about divergence of certain norms

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 (1975) and independently later in Holmer-Roudenko (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 in Chapter 7 of Sulem-Sulem (1998).