Mass concentration phenomena for the L²-critical nonlinear Schrödinger equation

In this paper, we show that any solution of the nonlinear Schrödinger equation iuₜ+Δ u± |u|⁴/Nu=0, which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on Bourgain’s (1998), 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 (1999). We also generalize to higher dimensions the results in Keraani (2006) and Merle and Vega (1998).