Mass Concentration Phenomena for the L^2-Critical Nonlinear Schr{\"o}dinger Equation

TL;DR

This paper proves that solutions to the L^2-critical nonlinear Schrödinger equation that blow up in finite time exhibit mass concentration near the blow-up, extending previous results to higher dimensions.

Contribution

It generalizes mass concentration results for finite-time blow-up solutions of the L^2-critical nonlinear Schrödinger equation to higher dimensions.

Findings

Mass concentration occurs near blow-up time.

Extension of previous 2D results to higher dimensions.

Utilizes Bourgain's method and generalized Strichartz inequalities.

Abstract

In this paper, we show that any solution of the nonlinear Schr{{\"o}}dinger equation $iu\_t+\Delta u\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgain's one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{MR1628235}.