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On sharpness of the local Kato-smoothing property for dispersive wave equations


Authors: Shu-Ming Sun, Emmanuel Trélat, Bing-Yu Zhang and Ning Zhong
Journal: Proc. Amer. Math. Soc. 145 (2017), 653-664
MSC (2010): Primary 35B65, 35Q53, 35Q55
DOI: https://doi.org/10.1090/proc/13286
Published electronically: August 5, 2016
MathSciNet review: 3577868
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Abstract: Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations

$\displaystyle w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb{R}^n, \ t\in \mathbb{R} , $

enjoy the local smoothing property

$\displaystyle q\in H^s (\mathbb{R}^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac {m-1}{2}}_{loc} \left (\mathbb{R}^n\right )\Big ) , $

where $ m$ is the order of the pseudo-differential operator $ P(D)$. This property, now called local Kato-smoothing, was first discovered by Kato for the KdV equation and implicitly shown later by Sjölin and Vega independently for the linear Schrödinger equation. In this paper, we show that the local Kato-smoothing property possessed by solutions of general dispersive equations in the 1D case is sharp, meaning that there exist initial data $ q\in H^s \left (\mathbb{R} \right )$ such that the corresponding solution $ w$ does not belong to the space $ L^2 \Big (-T,T; H^{s+\frac {m-1}{2} +\epsilon }_{loc} \left (\mathbb{R}\right )\Big )$ for any $ \epsilon >0$.

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Additional Information

Shu-Ming Sun
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
Email: sun@math.vt.edu

Emmanuel Trélat
Affiliation: Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, 4 place Jussieu, 75005, Paris, France
Email: emmanuel.trelat@upmc.fr

Bing-Yu Zhang
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221 – and – Yangtze Center of Mathematics, Sichuan University, Chengdu, People’s Republic of China
Email: zhangb@ucmail.uc.edu

Ning Zhong
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221
Email: zhongn@ucmail.uc.edu

DOI: https://doi.org/10.1090/proc/13286
Keywords: Local Kato smoothing property, dispersive wave equations, the KdV equation, the Schr\"odinger equation
Received by editor(s): March 27, 2016
Received by editor(s) in revised form: March 31, 2016
Published electronically: August 5, 2016
Additional Notes: The first author was partially supported by the National Science Foundation under grant No. DMS-1210979.
The third author was partially supported by a grant from the Simons Foundation (201615), and the NSF of China (11231007, 11571244).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society