The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions

By Monica Visan. 2005

294 citations recorded by the source corpus.

We obtain global well-posedness, scattering, and global $L^{\frac{2(n+2)}{n-2}}_{t,x}$ spacetime bounds for energy-space solutions to the energy-critical nonlinear Schrodinger equation in $\R_t\times \R^n_x$, $n\geq 5$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a53f5b9bca8f84eb66df2ba1420bcfeea7e0cdc

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On the global well-posedness of energy-critical Schr\"odinger equations in curved spaces (2010, methodology): “[43] R. Killip, B. Stovall and M. Visan, Scattering for the cubic Klein–Gordon equation in two space dimensions, Preprint (2010).”
Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “12 (Fractional chain rule for a Hölder continuous function, [104]).”
The focusing energy-critical Hartree equation (2009, methodology): “The modification can be made in the same spirit with [11] by Killip and Visan on energy-critical nonlinear Schrödinger equations or with [9] by Kenig and Merle on Ḣ 1 2 x -critical problem.”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “This work was extended to four dimensions in [73] and to dimensions greater than ﬁve in [98].”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, methodology): “Both [71] and the current paper build on techniques developed in order to treat the energy-critical NLS; see [6, 16, 35, 52, 64, 74, 75].”
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].”
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”
The cubic fourth-order Schrödinger equation (2008, methodology): “We use this to prove an interaction Morawetz estimate in Sections 7 and 8, following previous analysis from Colliander, Keel, Staffilani, Takaoka and Tao [7], Ryckman and Visan [32] and Visan [40].”