Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

By Terence Tao. 2004

159 citations recorded by the source corpus.

In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schrodinger equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions $n \geq 6$ some new technical difficulties arise because of the very low power of the non-linearity.

Subfields: Mathematical Physics, Rheumatology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7337620fd7f79d653eb42269f5c7e9de0b3fd122

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Citations in the corpus

The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, methodology): “For further developments, see [Colliander et al. 2008a; Ryckman and Visan 2007; Tao 2005; Visan 2006; 2007].”
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”
Global Well-Posedness for Energy Critical Fourth-Order Schrodinger Equations in the Radial Case (2007, methodology): “The theorem is proved in Section 8 following the strategy initiated in Bourgain [2] and developed in Tao [ 26 ].”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, methodology): “These ideas where pursued further in the work that followed, for example, [16, 54, 66 , 76, 77].”
Quintic NLS in the exterior of a strictly convex obstacle (2012, methodology): “The argument was modelled on that of Bourgain [8] and Tao [65], and correspondingly considered only spherically symmetric data.”
The focusing energy-critical Hartree equation (2009, methodology): “In the defocusing case, the global wellposedness and scattering was established by Bourgain [1], Grillakis [4] and Tao [18] for spherically symmetric initial data, and by Colliander, Keel, Staffilani, Takaoka and Tao [3] Ryckman and Visan [16] and Visan [21,22] for arbitrary initial data.”
Nonlinear Schrodinger Equations at Critical Regularity (2013, methodology): “Subsequent progress in the spherically symmetric case, including the treatment of higher dimensions, can be found in [89].”
On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations (2006, background): “Tao [17], settled the problem for arbitrary dimensions (with an improvement in the final bound due to a simplification of the argument), but again only for radially-symmetric data.”
Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, background): “See [10], [25], [37], and [29] for more work on the defocusing, energy critical initial value problem.”
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].”