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Bounds for KdV and the 1-d cubic NLS equation in rough function spaces. (English) Zbl 1319.35219
Summary: We consider the cubic nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time \(H^s\) bounds in terms of the \(H^s\) size of the initial data for \[ s\geq -\frac14 \] (joint work with D. Tataru [Int. Math. Res. Not. 2007, No. 16, Article ID rnm053, 36 p. (2007; Zbl 1169.35055); Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 6, 955–988 (2012; Zbl 1280.35137)]), and the solutions to KdV satisfy global a priori estimate in \(H^{-1}\) (joint work with T. Buckmaster [“The Korteweg-de-Vries equation at \(H^{-1}\) regularity”, arXiv:1112.4657]).
MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
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