A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

By Justin Holmer, Svetlana Roudenko. 2007

322 citations recorded by the source corpus.

We consider the problem of identifying sharp criteria under which radial H1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂tu + Δu + |u|2u = 0 scatter, i.e., approach the solution to a linear Schrödinger equation as t → ±∞. 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 u0 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 eitQ(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 eitQ(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 [17] in their study of the energy-critical NLS.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8adf6dbdbab30530e9559d4092c9866482f679ff

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Citations in the corpus

On well posedness for the inhomogeneous nonlinear Schr\"odinger equation (2016, methodology): “However, the abstract theory developed by Cazenave and later used by GenoudStuart [13] to show well-posedness for (1.1), does not give sufficient tools to study other interesting questions, for instance, scattering and blow up investigated by Kenig-Merle [21], Holmer-Roudenko-Duyckaerts [9] and othe”
Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications (2008, methodology): “For instance, it was applied to the “mass-critical” NLS in the radial case, in dimension d ≥ 2, in both focusing and defocusing cases, in works of Killip, Tao, Vişan and Zhang, [38], [21], [23], to the focusing cubic NLS for d = 3, by Holmer–Roudenko [13] and Duyckaerts–Holmer–Roudenko [6], to corot”
The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher (2008, background): “[10] T. Duyckaerts, J. Holmer, and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, prerint arXiv:0710.3630.”