Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 2 (2018), 383-438.

Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

Terence Tao

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Abstract

We consider the global regularity problem for defocusing nonlinear Schrödinger systems

i t + Δ u = ( m F ) ( u ) + G

on Galilean spacetime ×d, where the field u:1+dm is vector-valued, F:m 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:×dm 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 d2 or d3 and p1+4(d2), 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 d3 and p>1+4(d2). 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.

Article information

Source
Anal. PDE, Volume 11, Number 2 (2018), 383-438.

Dates
Received: 1 December 2016
Revised: 23 June 2017
Accepted: 5 September 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513774509

Digital Object Identifier
doi:10.2140/apde.2018.11.383

Mathematical Reviews number (MathSciNet)
MR3724492

Zentralblatt MATH identifier
1375.35435

Subjects
Primary: 35Q41: Time-dependent Schrödinger equations, Dirac equations

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

Citation

Tao, Terence. Finite time blowup for a supercritical defocusing nonlinear Schrödinger system. Anal. PDE 11 (2018), no. 2, 383--438. doi:10.2140/apde.2018.11.383. https://projecteuclid.org/euclid.apde/1513774509


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