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On the regularity of solutions to the $k$-generalized Korteweg-de Vries equation


Authors: C. E. Kenig, F. Linares, G. Ponce and L. Vega
Journal: Proc. Amer. Math. Soc. 146 (2018), 3759-3766
MSC (2010): Primary 35Q53; Secondary 35B05
DOI: https://doi.org/10.1090/proc/13506
Published electronically: June 11, 2018
MathSciNet review: 3825831
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Abstract: This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial datum is $u_0\in H^l((b,\infty ))$ for some $l\in \mathbb {Z}^+$ and $b\in \mathbb {R}$, then the corresponding solution $u(\cdot ,t)$ belongs to $H^l((\beta ,\infty ))$ for any $\beta \in \mathbb {R}$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $l>3/4$.


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Additional Information

C. E. Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
MR Author ID: 100230
Email: cek@math.uchicago.edu

F. Linares
Affiliation: IMPA, Instituto Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil
MR Author ID: 343833
Email: linares@impa.br

G. Ponce
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
MR Author ID: 204988
Email: ponce@math.ucsb.edu

L. Vega
Affiliation: UPV/EHU, Departamento de Matemáticas, Apto. 644, 48080 Bilbao, Spain – and – Basque Center for Applied Mathematics, E-48009 Bilbao, Spain
MR Author ID: 237776
Email: luis.vega@ehu.es

Keywords: Nonlinear dispersive equation, propagation of regularity
Received by editor(s): April 9, 2016
Received by editor(s) in revised form: April 29, 2016, and September 25, 2016
Published electronically: June 11, 2018
Additional Notes: The first author was supported by the NSF grant DMS-1265249.
The second author was supported by CNPq and FAPERJ/Brazil.
The fourth author was supported by ERCEA Advanced Grant 2014 669689 - HADE and by the MINECO project MTM2014-53850-P
Communicated by: Catherine Sulem
Article copyright: © Copyright 2018 American Mathematical Society