The cubic nonlinear Schr\"odinger equation in two dimensions with radial data

By Rowan Killip, Terence Tao, Monica Visan. 2007

251 citations recorded by the source corpus.

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d8f2394a6f6e9fac6d10dc4b12392103ba3df00

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In Mathematical Physics

Citations in the corpus

The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher (2008, methodology): “The approach we take here is inspired by [3], [23].”
Maximizers for the Strichartz Inequalities for the wave equation (2009, methodology): “[25] T. Tao, M. Visan, and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math.”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “[57, 61] showed that Conjecture 1.1 is true when the L 2 norm of the initial data is below the L 2 norm of the ground state for d ≥ 2 and u 0 radial.”
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 mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, methodology): “The argument mimics that in [ 26 ], though the non-polynomial nature of the nonlinearity introduces several technical complications.”
Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “24 (Three enemies: the mass-critical NLS, [43]).”
Some non-existence results for the semilinear Schrödinger equation without gauge invariance (2015, background): “In the defocusing case (F (z) = |z|z), it was proved in [11, 12, 13] that the equation (NLS) is globally well-posed in L(2)(R) for arbitrary L(2)(R) initial data, and the solution scatters to a free solution in L(2)(R) as t→ ±∞ (see also [36] for the radially symmetric case).”
Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008, background): “9See [ 19 ] for the analogous theory in the case of the mass-critical NLS; note”
Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1 (2008, background): “In recent years, the more delicate problems that correspond to critical power nonlinearities, both in Ḣ(1) and L(2), have also been considered, see [11, 24, 40, 21, 37, 45, 32, 42, 33].”
Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, methodology): “Acknowledgements The author would like to thank Monica Visan for suggesting the problem and for valuable comments and suggestions on the manuscript, as well as helpful discussions.”
Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, methodology): “Global existence results were proved in the energycritical regime, by Struwe [102], Grillakis [57, 58], Shatah-Struwe [94], Kapitanski [66], Bahouri-Shatah [3], Bahouri-Gérard [2], Bourgain [16], Nakanishi [88, 89], Tao [105], Kenig-Merle [70], Ryckman-Visan [91], Visan [109], Colliander-Keel-Staffi”
A refinement of the Strichartz inequality for the wave equation with applications (2012, unknown): “The profile decomposition has traditionally proven useful in the nonlinear theory (see for example [13–15], [16,17], [26], [28], [30–33], [34], [36], [37], [38], [39], [40–42], [48] or [60,61]).”