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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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