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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system
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by Chulkwang Kwak and Claudio Muñoz PDF
Trans. Amer. Math. Soc. 373 (2020), 1043-1107 Request permission

Abstract:

Consider the Hamiltonian $abcd$ system in one dimension, with data posed in the energy space $H^1\times H^1$. This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where $a,c<0$ and $b=d>0$. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this $2\times 2$ system was given in [J. Math. Pure Appl. (9) 127 (2019), 121–159] in a strongly dispersive regime, i.e., under essentially the conditions \[ b=d > \frac 29, \quad a,c<-\frac 1{18}. \] Additionally, decay was obtained inside a proper subset of the light cone $(-|t|,|t|)$. In this paper, we improve [J. Math. Pure Appl. (9) 127 (2019), 121–159] in three directions. First, we enlarge the set of parameters $(a,b,c,d)$ for which decay to zero is the only available option, considering now the so-called weakly dispersive regime $a,c\sim 0$: we prove decay if now \[ b=d > \frac 3{16}, \quad a,c<-\frac 1{48}. \] This result is sharp in the case where $a=c$, since for $a,c$ bigger, some $abcd$ linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form $|x|\sim |v|t$ for any $|v|<1$. This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small $abcd$ solutions in exterior regions $|x|\gg |t|$, also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced.
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Additional Information
  • Chulkwang Kwak
  • Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, 4860 Macul, Chile; and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju-si, Jeollabuk-do, 54896 Republic of Korea
  • MR Author ID: 1035460
  • Email: chkwak@mat.uc.cl
  • Claudio Muñoz
  • Affiliation: CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
  • MR Author ID: 806855
  • Email: cmunoz@dim.uchile.cl
  • Received by editor(s): May 14, 2019
  • Published electronically: October 18, 2019
  • Additional Notes: The first author was supported by FONDECYT Postdoctorado 2017 Proyecto No. 3170067 and project France-Chile ECOS-Sud C18E06.
    The second author’s work was partly funded by Chilean research grants FONDECYT 1150202, Fondecyt no. 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 1043-1107
  • MSC (2010): Primary 35Q35, 35Q51
  • DOI: https://doi.org/10.1090/tran/7944
  • MathSciNet review: 4068258