Minimal-mass blowup solutions of the mass-critical NLS

By Terence Tao, Monica Visan, Xiaoyi Zhang. 2006

159 citations recorded by the source corpus.

Abstract We consider the minimal mass m 0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|4/d u to blow up. If m 0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L 2-critical nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in for the defocusing NLS in three and higher dimensions with spherically symmetric data.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff2d7571e3b02cd568f727183bdf244b9ef6baf4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. B. Dodson
2. Aynur Bulut
3. Rowan Killip
4. A. Lawrie
5. Terence Tao
6. Monica Visan
7. M. Vi
8. Carlos E. Kenig

In Mathematical Physics

1. B. Dodson
2. Aynur Bulut
3. Rowan Killip
4. Monica Visan
5. A. Lawrie
6. Terence Tao
7. M. Vi
8. Carlos E. Kenig

Citations in the corpus

• Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “23 (Asymptotic orthogonality to free evolutions, [96]).”
• The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “By hypothesis and Definition 1.14, we see that for every ε > 0 there exists R > 0 such that ∫ |x|≥R |u(n)(0, x)|2 dx ≤ ε and ∫ |ξ|≥R |û(n)(0, ξ)|2 dξ ≤ ε for all n.”
• 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]).”
• Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, methodology): “This was proved in [90, 92], so the argument will merely be sketched here.”
• Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, methodology): “[26, 42] Let u : I × R → R be a solution to (NLW) with maximal interval of existence I which is almost periodic modulo symmetries.”
• Maximizers for the Strichartz Inequalities for the wave equation (2009, methodology): “[25] T. Tao, M. Visan, and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math.”
• 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].”
• Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d =3$ (2009, background): “Furthermore, [33] proved that for conjecture 1.”
• Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state (2011, unknown): “(1.22) Proof: See theorem 5.1 of [20] for d ≥ 3, theorem 5.2 of [19] for d = 2, and theorem 6.2 of [18] for d = 1.”
• Scattering for the radial 3D cubic wave equation (2014, methodology): “ a minimal non-scattering solution ~u(t) to (1.4) so that SC(AC,~u) fails, which enjoys certain compactness properties. We will state a refined version of this result below, and we refer the reader to [23, 44, 50, 51] for the details of the argument. As usual, the main ingredients are the linear and ”