Periodic-Coefficient Damping Estimates, and Stability of Large-Amplitude Roll Waves in Inclined Thin Film Flow

Rodrigues, Luis Miguel; Zumbrun, Kevin

doi:10.1137/15m1016242

Cited by 14 publications

(9 citation statements)

References 22 publications

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“…A perspective we did not investigate here would be the study of a perturbations around a periodic state. The periodicity property, used similarly as in [26], should enable us to overcome the previous difficulty and should give us the suitable gauge estimates.…”

Section: Discussionmentioning

confidence: 99%

On the well-posedness of a quasi-linear Korteweg-de Vries equation

Mietka¹

2017

Annales Mathématiques Blaise Pascal

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cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques Abstract The Korteweg-de Vries equation (KdV) and various generalized, most often semilinear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamiltonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major interest being the method used to get them. The goal is to make use of the structural properties of the equation, namely the skew-symmetry of the leading order term, and then to control subprincipal terms using suitable gauges as introduced by Lim & Ponce [22] and developed later by Kenig, Ponce & Vega [20] and S. Benzoni-Gavage, R. Danchin & S. Descombes [4]. The existence of a solution is obtained as a limit from regularized parabolic problems. Uniqueness and continuity with respect to the initial data are proven using a Bona-Smith regularization technique.This work has been supported by ANR project BoND (ANR-13-BS01-0009-01).

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Section: Discussionmentioning

confidence: 99%

On the well-posedness of a quasi-linear Korteweg-de Vries equation

Mietka¹

2017

Annales Mathématiques Blaise Pascal

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“…A perspective we did not investigate here would be the study of a perturbations around a periodic state. The periodicity property, used similarly as in [22], should enable us to overcome the previous difficulty and should give us the suitable gauge estimates.…”

Section: Discussionmentioning

confidence: 99%

On the well-posedness of a quasi-linear Korteweg-de Vries equation

Mietka¹

2016

Preprint

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The Korteweg-de Vries equation (KdV) and various generalized, most often semilinear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamiltonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major interest being the method used to get them. The goal is to make use of the structural properties of the equation, namely the skew-symmetry of the leading order term, and then to control subprincipal terms using suitable gauges as introduced by Lim

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“…Note that this occurs even when, as here, the relaxation is degenerate in the sense that it appears only in one equation of the system. In particular the condition on the index I introduced below is analogous to the averaged slope condition of the viscous case that was shown to always hold in [RZ16]. 6.1.…”

Section: High-frequency Analysismentioning

confidence: 99%

Spectral Stability of Inviscid Roll Waves

Johnson

Noble

Rodrigues

et al. 2018

Commun. Math. Phys.

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We carry out a systematic analytical and numerical study of spectral stability of discontinuous roll wave solutions of the inviscid Saint Venant equations, based on a periodic Evans-Lopatinsky determinant analogous to the periodic Evans function of Gardner in the (smooth) viscous case, obtaining a complete spectral stability diagram useful in hydraulic engineering and related applications. In particular, we obtain an explicit low-frequency stability boundary, which, moreover, matches closely with its (numerically-determined) counterpart in the viscous case. This is seen to be related to but not implied by the associated formal first-order Whitham modulation equations.

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Periodic-Coefficient Damping Estimates, and Stability of Large-Amplitude Roll Waves in Inclined Thin Film Flow

Cited by 14 publications

References 22 publications

On the well-posedness of a quasi-linear Korteweg-de Vries equation

On the well-posedness of a quasi-linear Korteweg-de Vries equation

On the well-posedness of a quasi-linear Korteweg-de Vries equation

Spectral Stability of Inviscid Roll Waves

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