TY - JOUR

T1 - Asymptotic dynamics for the small data weakly dispersive one-dimensional Hamiltonian ABCD system

AU - Kwak, Chulkwang

AU - Munoz, Claudio

N1 - Funding Information:
Received by the editors May 14, 2019. 2010 Mathematics Subject Classification. Primary 35Q35, 35Q51. 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.
Publisher Copyright:
© 2019 American Mathematical Society.

PY - 2020

Y1 - 2020

N2 - Consider the Hamiltonian abcd system in one dimension, with data posed in the energy space H1 × H1. 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 × 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 2 1 b = d > 9 , a, c < – 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 ∼ 0: we prove decay if now 1 b = d > 16 3 , a, c < – 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| ∼ |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| |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.

AB - Consider the Hamiltonian abcd system in one dimension, with data posed in the energy space H1 × H1. 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 × 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 2 1 b = d > 9 , a, c < – 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 ∼ 0: we prove decay if now 1 b = d > 16 3 , a, c < – 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| ∼ |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| |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.

UR - http://www.scopus.com/inward/record.url?scp=85080901934&partnerID=8YFLogxK

U2 - 10.1090/tran/7944

DO - 10.1090/tran/7944

M3 - Article

AN - SCOPUS:85080901934

VL - 373

SP - 1043

EP - 1107

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -