Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

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 $F (v) = 1 ∕ (p + 1) | v |^{p + 1}$ . It is well known that in the energy-subcritical and energy-critical cases when $d \leq 2$ or $d \geq 3$ and $p \leq 1 + 4 ∕ (d - 2)$ , one has global existence of smooth solutions from arbitrary smooth compactly supported initial data $u (0)$ and forcing term $G$ , at least in low dimensions. In this paper we study the supercritical case where $d \geq 3$ and $p > 1 + 4 ∕ (d - 2)$ . 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 $u (0)$ 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.

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Keywords

discretely self-similar blowup, finite time blowup, nonlinear Schrödinger equation

Mathematical Subject Classification 2010

Primary: 35Q41

Milestones

Received: 1 December 2016

Revised: 23 June 2017

Accepted: 5 September 2017

Published: 17 October 2017

Authors

Terence Tao
Department of Mathematics UCLA Los Angeles, CA United States

Vol. 11, No. 2, 2018

Terence Tao

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors