Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr

By I. Bejenaru, Terence Tao. 2005

212 citations recorded by the source corpus.

Abstract We establish that the quadratic non-linear Schrodinger equation iu t + u xx = u 2 , where u : R × R → C , is locally well-posed in H s ( R ) when s ⩾ - 1 and ill-posed when s - 1 . Previous work in [C. Kenig, G. Ponce, L. Vega, Quadratic forms for the 1-D semilinear Schrodinger equation, Trans. Amer. Math. Soc. 346 (1996) 3323–3353] had established local well-posedness for s > - 3 4 . The local well-posedness is achieved by an iteration using a modification of the standard X s , b spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1c19b218945a0c49ccfd5e4c5865b9e648f86f65

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Global well-posedness of Korteweg-de Vries equation in H^-3/4 (R) (2008, result): “After this paper was published, the author learned that Nobu Kishimoto [16] also obtained similar results with different resolution spaces by using some ideas in [1].”
Nonparametric Instrumental Variable Estimation Under Monotonicity (2015, methodology): “To this end, we introduce the concept of (local) quantitative well-posedness, which is inspired by the definitions in Bejenaru and Tao (2006), who studied the Cauchy problem for the quadratic non-linear Schrödinger equation.”
Invariance of the White Noise for KdV (2009, background): “Bejenaru-Tao [1], Kishimoto [17] in the context of NLS.”
Local well-posedness for quadratic nonlinear Schrödinger equations and the ``good'' Boussinesq equation (2010, methodology): “[13] T. Muramatu and S. Taoka, The initial value problem for the 1-”