Finite time blowup for a supercritical defocusing nonlinear Schr\"odinger system

TL;DR

This paper constructs explicit examples of finite time blowup solutions for supercritical defocusing nonlinear Schrödinger systems, showing that global regularity cannot be generally expected in these cases.

Contribution

It demonstrates finite time singularity formation in supercritical defocusing NLS systems with vector-valued fields, extending previous results from scalar equations.

Findings

Existence of smooth potentials leading to blowup

Finite time singularities in supercritical regimes

Solutions exhibit local discrete self-similarity

Abstract

We consider the global regularity problem for defocusing nonlinear Schr\"odinger systems $$ i \partial_t + \Delta u = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ 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 $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schr\"odinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$. In this paper we study the supercritical case where $d \geq 3$ and $p > 1 + \frac{4}{d-2}$. 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 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 Schr\"odinger systems. As in a previous paper of the author 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.