On sharpness of the local Kato-smoothing property for dispersive wave equations
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- by Shu-Ming Sun, Emmanuel Trélat, Bing-Yu Zhang and Ning Zhong PDF
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Abstract:
Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations \[ w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb {R}^n, \ t\in \mathbb {R} , \] enjoy the local smoothing property \[ q\in H^s (\mathbb {R}^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac {m-1}{2}}_{loc} \left (\mathbb {R}^n\right )\Big ) , \] 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 $\epsilon >0$.References
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Additional Information
- Shu-Ming Sun
- Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
- MR Author ID: 271223
- 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
- MR Author ID: 310235
- Email: zhangb@ucmail.uc.edu
- Ning Zhong
- Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221
- MR Author ID: 232808
- Email: zhongn@ucmail.uc.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 653-664
- MSC (2010): Primary 35B65, 35Q53, 35Q55
- DOI: https://doi.org/10.1090/proc/13286
- MathSciNet review: 3577868