Global Well-Posedness of the Kirchhoff Equation and Kirchhoff Systems

Matsuyama, Tokio; Ruzhansky, Michael

doi:10.1007/978-3-319-12148-2_5

Cited by 9 publications

(8 citation statements)

References 20 publications

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“…The Cauchy problem for the Kirchhoff equation has been extensively studied, starting from the pioneering paper of Bernstein [11]. Both local and global existence results have been established for initial data in Sobolev and analytic class, see [1], [2], [24], [25], [37], [44], [47] and the recent survey [45]. The existence of periodic solutions for the Kirchhoff equation has been proved by Baldi [3].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$

Montalto

2017

International Mathematics Research Notices

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We prove a reducibility result for a class of quasi-periodically forced linear wave equations on the d-dimensional torus T d of the formwhere the perturbation P(ωt) is a second order operator of the form P(ωt) = −a(ωt)∆ − R(ωt), the frequency ω ∈ R ν is in some Borel set of large Lebesgue measure, the function a : T ν → R (independent of the space variable) is sufficiently smooth and R(ωt) is a time-dependent finite rank operator. This is the first reducibility result for linear wave equations with unbounded perturbations on the higher dimensional torus T d . As a corollary, we get that the linearized Kirchhoff equation at a smooth and sufficiently small quasi-periodic function is reducible.Keywords: reducibility of linear wave equations on T d , KAM for PDEs, unbounded perturbations.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$

Montalto

2017

International Mathematics Research Notices

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“…For more details, generalizations and other open questions, we refer to Lions [30], to the surveys of Arosio [1], Spagnolo [33], Matsuyama and Ruzhansky [31], and to other references in our previous paper [4].…”

Section: Related Literaturementioning

confidence: 99%

On the normal form of the Kirchhoff equation

Baldi¹,

Haus²

2020

Preprint

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Consider the Kirchhoff equationIn a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

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“…On compact domains, dispersion, scattering and time-decay mechanisms are not available, and there are no results of global existence, nor of finite time blowup, for initial data (α, β) of Sobolev, or C ∞ , or Gevrey regularity. The local wellposedness in the Sobolev class H ) and other open questions, we refer to Lions [37] and the surveys of Arosio [2], Spagnolo [49], and Matsuyama and Ruzhansky [41].…”

Section: Related Literature and Open Questionsmentioning

confidence: 99%

On the existence time for the Kirchhoff equation with periodic boundary conditions

Baldi

Haus

2019

Nonlinearity

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We consider the Cauchy problem for the Kirchhoff equation on T d with initial data of small amplitude ε in Sobolev class. We prove a lower bound ε −4 for the existence time, which improves the bound ε −2 given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.

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Global Well-Posedness of the Kirchhoff Equation and Kirchhoff Systems

Cited by 9 publications

References 20 publications

A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$

A Reducibility Result for a Class of Linear Wave Equations on ${\mathbb T}^d$

On the normal form of the Kirchhoff equation

On the existence time for the Kirchhoff equation with periodic boundary conditions

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