Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

Abstract

We consider the global regularity problem for defocusing nonlinear Schrödinger systems

$$i{\partial }_t+\Delta u=\left({\nabla }_{{ℝ}^m}F\right)\left(u\right)+G$$

on Galilean spacetime $ℝ\times {ℝ}^d$, where the field $u:{ℝ}^{1+d}\to {ℂ}^m$ is vector-valued, $F:{ℂ}^m\to ℝ$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p>1$, and $G:ℝ\times {ℝ}^d\to {ℂ}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinger (NLS) equation, in which $m=1$ and $F\left(v\right)=1∕\left(p+1\right)|v{|}^{p+1}$. It is well known that in the energy-subcritical and energy-critical cases when $d\le 2$ or $d\ge 3$ and $p\le 1+4∕\left(d-2\right)$, one has global existence of smooth solutions from arbitrary smooth compactly supported initial data $u\left(0\right)$ and forcing term $G$, at least in low dimensions. In this paper we study the supercritical case where $d\ge 3$ and $p>1+4∕\left(d-2\right)$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$, positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth compactly supported choice of initial data $u\left(0\right)$ and forcing term $G$ for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schrödinger systems.

As in a previous paper of the author (Anal. PDE 9:8 (2016), 1999–2030) considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation.