A priori bounds and weak solutions for the nonlinear Schr

By Michael Christ, James E. Colliander, Terence Tao. 2006

72 citations recorded by the source corpus.

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrodinger equation on the real line are studied in Sobolev spaces Hs, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the Hs norm of the solution, in terms of the Hs norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in Hs.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d1d261670a99c1f0585d6ef9b44f65be6afbadf

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In Mathematical Physics

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

Citations in the corpus

• 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.”
• 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).”
• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces (2018, unknown): “In [12, 25, 26], Christ-Colliander-Tao and Koch-Tataru independently proved existence (without uniqueness) of solutions to the cubic NLS (1.”
• Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1 (2010, unknown): “By [17, 18] Conjecture 1.1 is sharp.”
• Global well-posedness of the KP-I initial-value problem in the energy space (2007, methodology): “uency and low modulation. To achieve this we still use Xs,btype structures for the spaces F1(T) and N1(T), but only on small, frequency dependent time intervals. 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 ”