## Compactness at blow-up time for $$L^2$$ solutions of the critical nonlinear Schrödinger equation in 2D.(English)Zbl 0913.35126

This interesting paper is devoted to the study of compactness at blow-up time for $$L^2$$-solutions of the critical nonlinear Schrödinger equation $u_t= (i/4\pi)\Delta u\pm| u|^2u\quad (x\in\mathbb{R}^2, t>0)$ with $$L^2(\mathbb{R}^2)$$ initial data. The authors obtain a compactness property for sequences of free solutions (satisfying the linear Schrödinger equation under the same initial data) which have big $$L^4$$ norm. The main result consists in the following statement: If there is a sequence $$\{u^n\}\in L^2$$ such that $$\|\exp\{it\Delta\} u^n\|^4_{L^4}\geq \alpha> 0$$ and $$\| u^n\|^2_{L_2}$$ is bounded, then there exists a subsequence also denoted $$\{u^n\}$$, $${\mathcal U}$$ with $$\|{\mathcal U}\|_{{\mathcal L}^\varepsilon}\geq\beta> 1$$, and $$\xi_n, x_n\in \mathbb{R}^2$$, $$\rho>0$$ such that $\widetilde u^n= \exp\{it_n\Delta\} \Biggl\{\exp\{2\pi i\xi_n\} \rho^{-1}_n u^n\Biggl({\cdot\over \rho_n}\Biggr)\Biggr\} (\cdot- x_n)$ converges to $${\mathcal U}$$ weakly in $$L^2(\mathbb{R}^2)$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs
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