Scattering in the Energy Space for Boussinesq Equations

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

doi:10.1007/s00220-018-3099-7

Cited by 15 publications

(18 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consider now the weight function φ(x) := sech 6 x in (4.9). Then, (4.10) leads to the estimate (see [34] for similar results)…”

Section: )mentioning

confidence: 66%

“…We conjecture that decay to zero in the energy space should hold in any compact set. See also the works [5,34,26,21] for decay in extended regions of space which profit of internal directions for decay.…”

Section: )mentioning

confidence: 99%

“…Moreover, the results proved in this note and in [22,23,28,32] apply to equations which have long range nonlinearities, as well as very low or null decay rates. See also [34,26] for other applications of this technique to the case of Boussinesq equations, and [25] for an application to the case of the BBM equation.…”

Section: )mentioning

confidence: 99%

See 2 more Smart Citations

Breathers and the Dynamics of Solutions in KdV Type Equations

Muñoz

Ponce

2018

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

In this paper our first aim is to identify a large class of non-linear functions f (·) for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or "small" breathers solutions. Also we prove that all small, uniformly in time L 1 ∩ H 1 bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order t 1/2 around any compact set in space. This set is included in the linearly dominated dispersive region x ≪ t. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime.

show abstract

“…Consider now the weight function φ(x) := sech 6 x in (4.9). Then, (4.10) leads to the estimate (see [34] for similar results)…”

Section: )mentioning

confidence: 66%

Section: )mentioning

confidence: 99%

See 1 more Smart Citation

Breathers and the Dynamics of Solutions in KdV Type Equations

Muñoz

Ponce

2018

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The second problem for which interesting conclusions can be stated is a classic one-wave fluid model. In [41], the authors considered extending the previous results to the case of the good Boussinesq model in 1+1 dimensions,…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Kwak¹,

Muñoz²,

Poblete³

et al. 2019

Journal de Mathématiques Pures et Appliquées

Self Cite

View full text Add to dashboard Cite

“…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%

Decay in the one dimensional generalized Improved Boussinesq equation

Maulén

Muñoz

2020

SN Partial Differ. Equ. Appl.

Self Cite

View full text Add to dashboard Cite

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

show abstract

Scattering in the Energy Space for Boussinesq Equations

Cited by 15 publications

References 29 publications

Breathers and the Dynamics of Solutions in KdV Type Equations

Breathers and the Dynamics of Solutions in KdV Type Equations

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

Decay in the one dimensional generalized Improved Boussinesq equation

Contact Info

Product

Resources

About