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

Alejo, Miguel A.; Cortez, Manuel Fernando; Kwak, Chulkwang; Muñoz, Claudio

doi:10.1093/imrn/rnz038

Cited by 13 publications

(11 citation statements)

References 77 publications

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“…Moreover, there exist constant C 0 ą 0 and an increasing sequence of times t n Ñ `8 such that The previous result holds for arbitrarily large data in L 2 , despite the fact that 2D ZK is scattering critical (the standard scattering trick is uB x u " 1 t u, see Faminskii [10] for required linear decay estimates). We also present in (1.4) a mild decay rate valid along a sequence of times growing to infinity.…”

Section: Introduction and Main Resultsmentioning

confidence: 85%

“…(1.1). Recall that 1 3 ă r ă 3, 0 ď b ă 2 3`r and 0 ď br ă 2r 3`r . This set corresponds to the region of the plane where Theorem 1.2 holds.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This set corresponds to the region of the plane where Theorem 1.2 holds. Some important limiting cases are r « 1 3 , for which b ă 3 5 and br ă 1 5 ; and r « 3, for which b ă 1 3 and br ă 1. Here x " t represents the soliton region.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On local energy decay for large solutions of the Zakharov-Kuznetsov equation

Mendez¹,

Muñoz²,

Poblete³

et al. 2020

Preprint

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We consider the Zakharov-Kutznesov (ZK) equation posed in R d , with d " 2 and 3. Both equations are globally well-posed in L 2 pR d q. In this paper, we prove local energy decay of global solutions: if uptq is a solution to ZK with data in L 2 pR d q, then lim inffor suitable regions of space Ω d ptq Ď R d around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for H 1 pR d q solutions. As a byproduct, our results extend decay properties for KdV and quartic KdV equations proved by Gustavo Ponce and the second author. Sequential rates of decay and other strong decay results are also provided as well.

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Section: Introduction and Main Resultsmentioning

confidence: 85%

“…(1.1). Recall that 1 3 ă r ă 3, 0 ď b ă 2 3`r and 0 ď br ă 2r 3`r . This set corresponds to the region of the plane where Theorem 1.2 holds.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On local energy decay for large solutions of the Zakharov-Kuznetsov equation

Mendez¹,

Muñoz²,

Poblete³

et al. 2020

Preprint

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“…The proof Theorem 1.1 follows the introduction of a new virial identity, in the spirit of the previous results by Martel and Merle [20,21] in the gKdV case, and [12,1] in the BBM case. Note however that in those cases the functional involved is related to the mass (L 2 norm) of the solution.…”

Section: 2mentioning

confidence: 89%

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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“…More precisely, the proofs are based in a series of localized virial type arguments, similar to the ones used in [1,2,18,19,17,25,27]. In our case, we will use a combination of virials to obtain the integrability in time of the L 2 ˆL2 -norm of pφptq ´Q, p1 ´γB 2…”

Section: ´B2mentioning

confidence: 99%

Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation

Maulén¹

2021

Preprint

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We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space H 1 ˆL2 . This model has solitary waves with speeds ´1 ă c ă 1. When |c| approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space. Contents 1. Introduction 1.1. Setting 1.2. Standing waves 1.3. Main results 1.4. Idea of the proof Organization of this paper Acknowledgments 2. A virial identity for the (gGB) system 2.1. Decomposition of the solution in a vicinity of the soliton 2.2. Notation for virial argument 2.3. Virial estimate 2.4. End of Proposition 2.1 3. Transformed problem and second virial estimates 3.1. The transformed problem 3.2. Virial functional for the transformed problem 3.3. Proof of Proposition 3.1: first computations 3.4. First technical estimates 3.5. Controlling error and nonlinear terms 3.6. End of proof of Proposition 3.1 4. Gain of derivatives via transfer estimates 4.1. A virial estimate related to Mptq 4.2. Second set of technical estimates 4.3. Start of proof of Proposition 4.2 4.4. Controlling error terms 4.5. Controlling nonlinear terms 4.6. End of proof Proposition 4.2 5. A second transfer estimate 5.1. A virial identity for N ptq 5.2. Start of proof of Proposition 5.2 5.3. Control of nonlinear terms

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

Cited by 13 publications

References 77 publications

On local energy decay for large solutions of the Zakharov-Kuznetsov equation

On local energy decay for large solutions of the Zakharov-Kuznetsov equation

Decay in the one dimensional generalized Improved Boussinesq equation

Asymptotic stability manifolds for solitons in the generalized Good Boussinesq equation

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