Global existence versus finite time blowup dichotomy for the system of nonlinear Schr\"odinger equations

TL;DR

This paper develops a profile decomposition for a kinetic energy inequality, studies extremizers, and applies these results to establish a dichotomy between global existence and finite-time blowup for a multi-particle nonlinear Schrödinger system.

Contribution

It introduces a concentration-compactness approach for operator inequalities and extends the global existence versus blowup dichotomy to multi-particle systems.

Findings

Constructed extremizers for the kinetic energy inequality.

Derived properties of extremizers, including Euler-Lagrange equations.

Established a global existence versus blowup dichotomy for multi-particle nonlinear Schrödinger systems.

Abstract

We construct an extremizer for the kinetic energy inequality (except the endpoint cases) developing the concentration-compactness technique for operator valued inequality in the formulation of the profile decomposition. Moreover, we investigate the properties of the extremizer, such as the system of Euler-Lagrange equations, regularity and summability. As an application, we study a dynamical consequence of a system of nonlinear Schr\"odinger equations with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal. Using the critical element of the kinetic energy inequality, we establish a global existence versus finite time blowup dichotomy. This result extends the single particle result of Holmer and Roudenko to infinitely many particles system.