Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation

By Pascal B'egout, Ana Vargas. 2007

162 citations recorded by the source corpus.

In this paper, we show that any solution of the nonlinear Schr{o}dinger equation $iu_t+\Delta u\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgain's one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{MR1628235}.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/405b585ffe456a4dcd3e44137e301196a1399ca6

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. Shuanglin Shao
3. Rowan Killip
4. Xiaoyi Zhang
5. J. Ramos
6. Terence Tao
7. Monica Visan
8. Plamen Stefanov

In Mathematical Physics

1. B. Dodson
2. Rowan Killip
3. Shuanglin Shao
4. Terence Tao
5. Monica Visan
6. Carlos E. Kenig
7. Xiaoyi Zhang
8. M. Vi

Citations in the corpus

• The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “By the Fundamental Theorem of Calculus and Hölder’s inequality, |g(r)|2 = ∣∣∣ ∫ ∞ r g(ρ)g′(ρ) dρ ∣∣∣ ≤ r1−d ∣∣∣ ∫ ∞ r g(ρ)g′(ρ)ρd−1dρ ∣∣∣ ≤ r1−d‖g‖L2(ρd−1 dρ)‖g′‖L2(ρd−1 dρ) for any Schwartz function g : [0,∞) → C.”
• Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation (2009, methodology): “To begin, we recall the profile decomposition result in [2] in the notation of the symmetry group which preserves the mass and the Strichartz inequalities.”
• The geodesic X-ray transform with fold caustics (2010, background): “A powerful result [Shao 2009] that leads easily to existence of extremizers is the profile decomposition; see [Bégout and Vargas 2007].”
• Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, background): “[3, 66, 92] extended this result to dimensions d ≥ 1.”
• Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions (2007, background): “, [33], [7], [6], [2], [20], [21], [5], [18], [1], [29], [32]).”
• Quintic NLS in the exterior of a strictly convex obstacle (2012, background): “In the Euclidean setting, existence was first proved by Keraani [43] for the (particularly difficult) mass-critical NLS; see also [3, 13].”
• Existence of Extremals for a Fourier Restriction Inequality (2010, background): “A powerful result [Shao 2009] that leads easily to existence of extremizers is the profile decomposition; see [Bégout and Vargas 2007].”
• Nonlinear Schrodinger Equations at Critical Regularity (2013, background): “Once again we turn to [4] for advice, this time, to the proof of their Theorem 1.”
• A refinement of the Strichartz inequality for the wave equation with applications (2012, methodology): “3 in [2] or the forthcoming Lemma 2.”