Ground state mass concentration in the $L^2$-critical nonlinear Schrödinger equation below $H^1$

We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schrödinger equation on ${\mathbb R}^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in $L^2$ are known to concentrate at least a small amount of mass. In this paper we consider the intermediate case of blowup solutions from initial data in $H^s$, with $1 > s > s_Q$, where $s_Q = \frac{1}{5} + \frac{1}{5} \sqrt{11}$. Our main result is that such solutions, when radially symmetric, concentrate at least the mass of the ground state at the origin at blowup time.

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Published 1 January 2005