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

Abstract

We consider finite time blowup solutions of the L2-critical cubic focusing nonlinear Schrodinger equation on R 2. Such functions, when in H1, are known to concentrate a fixed L2-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in L2 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 Hs, with 1 > s > sQ, where sQ = 1 5 + 1 5 √ 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.