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Global well-posedness for the \(L^2\) critical nonlinear Schrödinger equation in higher dimensions. (English) Zbl 1152.35106

From the text: We study the \(L^2\) critical defocusing Cauchy problem \[ iu_t+ \Delta u-|u|^{{4\over n}} u= 0,\quad x\in\mathbb{R}^n,\;t\geq 0, \]
\[ u(x,0= u_0(x)\in H^s(\mathbb{R}^n),\quad n\geq 3, \] where \(u(x, t)\) is a complex-valued function in space-time \(\mathbb{R}^n\times \mathbb{R}^+\). Here \(H^s(\mathbb{R}^n)\) denotes the usual inhomogeneous Sobolev space.
We show that the problem is globally well-posed in \(H^s(\mathbb{R}^n)\) when \(1> s> {\sqrt{7}-1\over 3}\) for \(n= 3\), and when \(1> s> {-(n-2)+ \sqrt{(n-2)^2+ 8(n- 2)}\over 4}\) for \(n\geq 4\). We use the “\(I\)-method” combined with a local in time Morawetz estimate.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
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