A Priori Bounds for the 1D Cubic NLS in Negative Sobolev Spaces

By Herbert Koch, D. Tataru. 2006

181 citations recorded by the source corpus.

Author(s): Koch, Herbert; Tataru, Daniel | Abstract: We consider the cubic Nonlinear Schrodinger Equation in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time H^s bounds in terms of the H^s size of the initial data for s greater than or equal to -1/6.

Subfields: Control and Systems Engineering, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/f6ba563615841a9904e06dcdbe8cb018d4d717a5

How researchers see this paper

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

In Control and Systems Engineering

1. Carlos E. Kenig
2. A. Ionescu
3. Zihua Guo
4. Didier Pilod
5. Luc Molinet
6. Jacob Sterbenz
7. B. Dodson
8. D. Tataru

In Mathematical Physics

1. Carlos E. Kenig
2. Didier Pilod
3. Zihua Guo
4. A. Ionescu
5. Luc Molinet
6. B. Dodson
7. D. Tataru
8. Jacob Sterbenz

In Mechanics of Materials

1. B. Dodson
2. Carlos E. Kenig
3. Tadahiro Oh
4. A. Ionescu
5. Didier Pilod
6. Yuzhao Wang
7. Zihua Guo
8. Luc Molinet

Citations in the corpus

• Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, methodology): “They have been successfully used for instance in [27], [28], [3], [10].”
• Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d=2$ (2010, methodology): “18) Function Spaces: We utilize the function spaces used in [24].”
• A priori bounds and weak solutions for the nonlinear Schr (2006, unknown): “Koch and Tataru [10] have obtained the same result in the larger range s − 16 .”
• Sharp ill-posedness results for the KdV and mKdV equations on the torus (2011, background): “Recall that these spaces are constructed by localizing in time the Bourgain spaces with a localization in time that depends inversely on the space frequencies of the functions (see [14] and [5] for previous works in this direction).”
• Well-posedness for the fifth-order KdV equation in the energy space (2012, methodology): “We refer to [4], [23] for previous works using similar spaces to prove a priori bounds for the 1D cubic NLS at low regularity and also to [11], [12], [29] for applications to other dispersive equations.”
• WIENER RANDOMIZATION ON UNBOUNDED DOMAINS AND AN APPLICATION TO ALMOST SURE WELL-POSEDNESS OF NLS (2014, background): “5 can also be proven with the variants of the X-spaces adapted to the U- and V -spaces introduced by Koch, Tataru, and their collaborators [36, 29, 30].”
• Global well-posedness of the KP-I initial-value problem in the energy space (2007, methodology): “A similar method was used recently in [5] and [ 13 ] to prove a-priori bounds for the 1-d cubic nonlinear Schr¨odinger equation in negative Sobolev spaces.”
• On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, background): “1 is based on the fixed point argument involving the variants of the X-spaces adapted to the U- and V -spaces introduced by Koch, Tataru, and their collaborators [29, 30, 41].”
• Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, methodology): “[62, 63] applied these spaces to nonlinear Schrödinger problems.”
• Strichartz estimates for Schrödinger equations on irrational tori (2013, methodology): “Furthermore, by applying the well-posedness theory involving the U- and V -spaces developed by Tataru, Koch, and their collaborators [18, 21, 22, 30], we prove some critical local well-posedness.”
• Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, background): “These spaces were first introduced in unpublished work of the second author, and have proved their usefulness as scale invariant substitutes of X 1 2 type spaces in several problems, see [11], [12], [5], [2].”
• Improvement of the energy method for strongly non resonant dispersive equations and applications (2014, unknown): “This type of method was first introduced by Koch and Tzvetkov [19] (see also [17] for some improvements) in the framework of Strichartz’s spaces and then by Koch and Tataru [18] (see also [13]) in the framework of Bourgain’s spaces.”
• Probabilistic well-posedness for the cubic wave equation (2011, unknown): “Notice still that in the range studied by Colliander–Oh, the (Wick-ordered) NLS appears not to be ill-posed in the sense of Theorem 1 (see the estimates by Koch–Tataru [20] on the line and the work announced by Christ–Holmer–Tataru [13] on the torus) and it might very well happen that a deterministi”