The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher

By Rowan Killip, Monica Visan. 2008

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We consider the focusing energy-critical nonlinear Schr\"odinger equation $iu_t+\Delta u = - |u|^{\frac4{d-2}}u$ in dimensions $d\geq 5$. We prove that if a maximal-lifespan solution $u:I\times\R^d\to \C$ obeys $\sup_{t\in I}\|\nabla u(t)\|_2<\|\nabla W\|_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle in \cite{Evian, kenig-merle} for spherically symmetric initial data and dimensions $d=3,4,5$.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/821bb02c8628e188e6f303601f353edbd06c8190

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In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Global well-posedness and scattering for nonlinear Schr\"odinger equations with combined nonlinearities in the radial case (2014, methodology): “After giving existence of the ground state based on the idea in [7] and [13], they showed a suﬃcient and necessary condition for the scattering in the spirit of [15] by using the proﬁle decomposition in ˙ H 1 ( R d ) and the global wellposedness and scattering result in [18].”
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].”
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 ].”
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]).”
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”
Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “11) is negative for u0, then the solution will blow up in finite time; for complete details see [44].”
Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions (2007, methodology): “This is because the Morawetz inequality is now subcritical instead of supercritical, and to prevent evacuation of mass to low frequencies we use the conservation of the subcritical energy (in contrast to [9], [23], [ 37 ], where the conservation of the supercritical mass is used to prevent evacuatio”
Some non-existence results for the semilinear Schrödinger equation without gauge invariance (2015, background): “The scattering results also were obtained for H (1)-solution, (see [32, 37]).”
The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher (2007, methodology): “For further developments, see [14, 32, 37, 43, 44 ].”
Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “Global well-posedness and scattering for the energy-critical defocusing nonradial Schr¨odinger equation [24, 59, 73, 97, 98, 99] and for the energy-critical focusing problem [33, 51, 58, 83] is now complete except for the focusing problem when d = 3 with nonradial data.”
Scattering for the radial 3D cubic wave equation (2014, methodology): “This technique was developed by Tao in [47] and utilized in the Kenig-Merle framework for nonlinear Schrödinger problems by Killip and Visan [27, 28, 30], and for semilinear wave equations in [29, 5, 6].”
Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “This result was for spherically symmetric data and dimensions 3 ≤ d ≤ 5; currently, the analogous result for general data is only known in dimensions five and higher [46].”
Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, methodology): “The version that we use here was obtained by Killip and Visan in [24].”