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Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions. (English) Zbl 1187.35246

This paper deals with strong \(L_x^2(\mathbb R^n)\)-solutions to the mass-critical (or pseudoconformal) defocusing nonlinear Schrödinger equation \(iu_t+\Delta u=|u|^{4/n}u\) \((*)\). The aim of the paper is to verify the special case of the following conjecture in the case where \(n\geq 3\) and \(u_0\) (and \(u_{\pm}\)) are restricted to be spherically symmetric:
Conjecture (Global existence and scattering): Given any finite-mass initial data \(u_0\in L_x^2(\mathbb R^n)\), there exists a unique global solution \(u\in C_t^0L_x^2(\mathbb R\times \mathbb R^n)\cup L_{t,x}^{2(n+2)/n}(\mathbb R\times\mathbb R^n)\) to the equation \((*)\) with \(u(0)=u_0\). Furthermore, there exist \(u_{\pm}\in L_x^2(\mathbb R^n)\) such that \(\lim_{t\to\pm\infty}\|u(t)-e^{it\Delta}u_{\pm}\|_{L_x^2(\mathbb R^n)}= 0\), and the maps \(u_0\mapsto u_{\pm}\) are homeomorphisms on \(L_x^2(\mathbb R^n)\). The authors borrow from the treatment of the energy-critical problem. They combine the frequency-localized Morawetz inequality approach used by Bourgain and others, and the more recent approach by C. E. Kenig and F. Merle [Invent. Math. 166, No. 3, 645–675 (2006; Zbl 1115.35125)], which considers an actual blowup solution (rather than an almost blowup solution). The advantage of this combined approach is that there is enough regularity to use the classical energy conservation law. The assumptions on the dimension (\(n\geq 3)\) and of spherical symmetry are used in an essential way in the proof.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B44 Blow-up in context of PDEs

Citations:

Zbl 1115.35125
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References:

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