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Ground state mass concentration in the \(L^2\)-critical nonlinear Schrödinger equation below \(H^1\). (English) Zbl 1084.35088

The authors consider finite time blow-up solutions to the initial value problem for the two-dimensional cubic \(L^2\)-critical focusing nonlinear Schrödinger equation \(i u_t+\Delta u+| u|^2 u=0, u(0,x)=u_{0}(x), x\in \mathbb{R}^2\) with the property that both the equation and the \(L^2\)-norm of the solution are invariant under the scaling transformation \(u(t,x)\rightarrow u_{\lambda}(t,x)=\lambda u(\lambda^2 t,\lambda x)\). This problem is locally well-posed for initial data in \(H^s\). The purpose of the article is to address the phenomenon of mass concentration at the point of blow-up in the spaces \(H^s,s<1\). For the intermediate case of blow-up solutions from initial data in \(H^s\) with \(1>s>s_{Q}=\frac{1}{5}+\frac{1}{5}\sqrt{11}\) it is shown that such radially symmetric solutions concentrate at least the mass of the ground state at the origin at blow-up time.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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