A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Justin Holmer; Svetlana Roudenko

doi:10.1007/s00220-008-0529-y

paper

A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

arXiv:math/0703235 · doi:10.1007/s00220-008-0529-y

Abstract

We consider the problem of identifying sharp criteria under which radial $H^{1}$ (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) $i \partial_{t} u + Δ u + ∣ u ∣^{2} u = 0$ scatter, i.e. approach the solution to a linear Schrödinger equation as $t \to \pm \infty$ . The criteria is expressed in terms of the scale-invariant quantities $∥ u_{0} ∥_{L^{2}} ∥\nabla u_{0} ∥_{L^{2}}$ and $M [u] E [u]$ , where $u_{0}$ denotes the initial data, and $M [u]$ and $E [u]$ denote the (conserved in time) mass and energy of the corresponding solution $u (t)$ . The focusing NLS possesses a soliton solution $e^{i t} Q (x)$ , where $Q$ is the ground-state solution to a nonlinear elliptic equation, and we prove that if $M [u] E [u] < M [Q] E [Q]$ and $∥ u_{0} ∥_{L^{2}} ∥\nabla u_{0} ∥_{L^{2}} < ∥ Q ∥_{L^{2}} ∥\nabla Q ∥_{L^{2}}$ , then the solution $u (t)$ is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution $e^{i t} Q (x)$ , for which equality in these conditions is obtained, is global but does not scatter. We further show that if $M [u] E [u] < M [Q] E [Q]$ and $∥ u_{0} ∥_{L^{2}} ∥\nabla u_{0} ∥_{L^{2}} > ∥ Q ∥_{L^{2}} ∥\nabla Q ∥_{L^{2}}$ , then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in their study of the energy-critical NLS.

A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Abstract

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