Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions

By Terence Tao, Monica Visan, Xiaoyi Zhang. 2007

166 citations recorded by the source corpus.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0c6f8cbf616c370a87748e1bb115908b437a5543

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Citations in the corpus

Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “The parameter N(t) measures the frequency scale of the solution at time t, and 1/N(t) measures the spatial scale; see [43, 96, 97] for further discussion.”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, methodology): “Both [ 73 ] and the current paper build on techniques developed in order to treat the energy-critical NLS; see [6, 16, 35, 54, 66, 76, 77].”
The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, methodology): “Remarks. 1. The parameter N(t) measures the frequency scale of the solution at time t, and 1/N(t) measures the spatial scale; see [39, 40 ] for further discussion.”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, background): “13) Proof: See [33] and section four of [31].”
The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality (2008, methodology): “It was also used to show the existence of minimal mass or energy blow-up solutions for the Schrödinger or wave equations at critical regularity, which is an important step in establishing the global well-posedness and scattering results for such equations, see [11], [12], [16], [30], [17].”
Why are solitons stable (2008, unknown): “global existence) for critical equations in the defocusing case, or in focusing cases in which the solution is “smaller” in mass or energy than that of the ground state. See [28], [ 81 ], [30]”
Scattering for the radial 3D cubic wave equation (2014, methodology): “SCATTERING FOR THE RADIAL 3D CUBIC WAVE EQUATION 473 Lemma 2.2 (radial Sobolev embedding [Tao et al. 2007, Corollary A.3]).”
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”
An alternative approach to regularity for the Navier–Stokes equations in critical spaces (2009, background): “Introduction In recent studies, the idea of establishing the existence of so-called “critical elements” (or the earlier “minimal blow-up solutions”) has led to significant progress in the theory of “critical” dispersive and hyperbolic equations such as the energy-critical nonlinear Schrödinger equat”
Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1 (2008, background): “has been proved for σ = 2/d under the radial symmetry assumption [34, 43 ].”
Nonlinear Schrodinger equation on real hyperbolic spaces (2008, methodology): “Global existence for arbitrary data in the critical case remains an open problem, although several results are available (Bourgain [5], Tao, Visan & Zhang [ 19 ], ...).”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “This was proved in [90, 92], so the argument will merely be sketched here.”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, background): “13) Proof: See [32] and section four of [30].”
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”
Minimal-mass blowup solutions of the mass-critical NLS (2006, methodology): “In a sequel to this paper, [ 38 ], we shall use Corollary 1.20 (or more precisely, the analogue of this corollary for spherically symmetric solutions, see Section 7) to establish global well-posedness and scattering for the defocusing NLS for spherically symmetric L2x(Rd) solutions in dimensions d ≥”