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Transactions of the American Mathematical Society

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Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

Authors: Chulkwang Kwak and Claudio Muñoz
Journal: Trans. Amer. Math. Soc. 373 (2020), 1043-1107
MSC (2010): Primary 35Q35, 35Q51
Published electronically: October 18, 2019
MathSciNet review: 4068258
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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

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

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.
Article copyright: © Copyright 2019 American Mathematical Society