Mass concentration phenomena for the 𝐿²-critical nonlinear Schrödinger equation

Bégout, Pascal; Vargas, Ana

doi:10.1090/S0002-9947-07-04250-X

Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation
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by Pascal Bégout and Ana Vargas PDF

Trans. Amer. Math. Soc. 359 (2007), 5257-5282 Request permission

Abstract:

In this paper, we show that any solution of the nonlinear Schrödinger 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 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).

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