The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher

By Rowan Killip, Monica Visan, Xiaoyi Zhang. 2007

172 citations recorded by the source corpus.

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/78fcd92682e3a2bd2f9352dc3f37cb7b38c9b848

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In Mathematical Physics

1. B. Dodson
2. Carlos E. Kenig
3. F. Merle
4. Rowan Killip
5. Aynur Bulut
6. M. Vi
7. G. Koch
8. Vedran Sohinger

Citations in the corpus

• Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, methodology): “To deal with the case when ∫∞ 0 N(t) 3dt < ∞, we use a method similar to the method used in [21], [32], and [23].”
• 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 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.”
• Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, methodology): “This method is similar to the method used in [21], [31], and [22].”
• 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”
• 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): “Recall that in the mass-critical case, the spherically symmetric soliton and cascade were shown to enjoy such additional regularity in [43, 46] for d ≥ 2.”