The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space

Kwak, Chulkwang; Muñoz, Claudio; Poblete, Felipe; Pozo, Juan C.

doi:10.1016/j.matpur.2018.08.005

Cited by 25 publications

(23 citation statements)

References 46 publications

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“…Note however that u seems not locally L 2 integrable in time. However, (1.14) shows that this norm indeed decays to zero in time (even if it is not integrable in time The techniques that we use to prove Theorem 1.2 are not new, and have been used to show decay for the Born-Infeld equation [2], the good Boussinesq system [24], the Benjamin-Bona-Mahony (BBM) equation [12], and more recently in the more complex abcd Boussinesq system [14,13]. In all these works, suitable virial functionals were constructed to show decay to zero in compact/not compact regions of space.…”

Section: 2mentioning

confidence: 99%

“…Controlling this last term requires some work. Indeed, we shall need the following definition (see [15,14,12] and references therein for more details)…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

“…Now we need the following result about equivalence of norms in terms of f and u. 12,14]). Let f be as in (3.2).…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

“…In what follows, we need the following auxiliary result. Lemma 3.4 ([15], see also [14]). The operator (1 − ∂ 2 x ) −1 satisfies the comparison priciple:…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

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Decay in the one dimensional generalized Improved Boussinesq equation

Maulén

Muñoz

2020

SN Partial Differ. Equ. Appl.

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We consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. This equation has been extensively studied in the literature, describing plenty of interesting behavior, such as global existence in the space H 1 × H 2 , existence of super luminal solitons, and lack of a standard stability method to describe perturbations of solitons. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity p is sufficiently large. In this paper we remove that condition on the power p and prove decay to zero in terms of the energy space norm L 2 × H 1 , for any p > 1, in two almost complementary regimes: (i) outside the light cone for all small, bounded in time H 1 × H 2 solutions, and (ii) decay on compact sets of arbitrarily large bounded in time H 1 × H 2 solutions. The proof consists in finding two new virial type estimates, one for the exterior cone problem based in the energy of the solution, and a more subtle virial identity for the interior cone problem, based in a modification of the momentum.Here the plus sign denotes the good Boussinesq system, which is locally and globally wellposed in standard Sobolev spaces [8,9], and the minus sign represents the "bad" equation originally derived by Boussinesq [4], which is strongly linearly ill-posed. Precisely, motivated Key words and phrases. Improved Boussinesq, decay, virial. (Ch.M.) Partially funded by Chilean research grants FONDECYT 1150202 and CONICYT PFCHA/DOCTORADO NACIONAL/2016-21160593. (Cl.M.) Partially funded by Chilean research grants FONDECYT 1150202 and 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. Part of this work was done while Cl.M. was visiting the CMLS at

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Section: 2mentioning

confidence: 99%

“…Controlling this last term requires some work. Indeed, we shall need the following definition (see [15,14,12] and references therein for more details)…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

“…Now we need the following result about equivalence of norms in terms of f and u. 12,14]). Let f be as in (3.2).…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

“…In what follows, we need the following auxiliary result. Lemma 3.4 ([15], see also [14]). The operator (1 − ∂ 2 x ) −1 satisfies the comparison priciple:…”

Section: Decay In Exterior Light Cones Proof Of Theorem 11mentioning

confidence: 99%

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Decay in the one dimensional generalized Improved Boussinesq equation

Maulén

Muñoz

2020

SN Partial Differ. Equ. Appl.

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“…In this paper, to prove Theorem 1.2 along increasing in time spatial intervals, we will adapt the ideas of [74] to the CH and DP cases. The novelty here is the nonlocal character of CH and DP, which makes the proofs slightly different in nature; in particular, we will need some of the estimates and properties proved in [54]. There is also in CH and DP some absence of Kato smoothing properties for the second derivatives of the solution.…”

Section: Solitons and Peakonsmentioning

confidence: 99%

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

Alejo

Cortez

Kwak

et al. 2019

International Mathematics Research Notices

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In this paper we consider globally defined solutions of Camassa-Holm (CH) type equations outside the well-known nonzero speed, peakon region. These equations include the standard CH and Degasperis-Procesi (DP) equations, as well as nonintegrable generalizations such as the b-family, elastic rod and BBM equations. Having globally defined solutions for these models, we introduce the notion of zero-speed and breather solutions, i.e., solutions that do not decay to zero as t → +∞ on compact intervals of space. We prove that, under suitable decay assumptions, such solutions do not exist because the identically zero solution is the global attractor of the dynamics, at least in a spatial interval of size |x| t 1/2− as t → +∞. As a consequence, we also show scattering and decay in CH type equations with long range nonlinearities. Our proof relies in the introduction of suitable Virial functionalsà la Martel-Merle in the spirit of the works [74,75] and [50] adapted to CH, DP and BBM type dynamics, one of them placed in L 1x , and a second one in the energy space H 1 x . Both functionals combined lead to local in space decay to zero in |x| t 1/2− as t → +∞. Our methods do not rely on the integrable character of the equation, applying to other nonintegrable families of CH type equations as well.

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Extended Decay Properties for Generalized BBM Equation

Kwak¹,

Muñoz

2019

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering

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The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space

Cited by 25 publications

References 46 publications

Decay in the one dimensional generalized Improved Boussinesq equation

Decay in the one dimensional generalized Improved Boussinesq equation

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

Extended Decay Properties for Generalized BBM Equation

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