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

Mietka, Colin

doi:10.5802/ambp.365

Cited by 5 publications

(11 citation statements)

References 25 publications

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We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous result in [16], generalising the considered class of equations and improving the regularity assumption on the initial data.

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confidence: 67%

On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations

Iandoli

2022

Preprint

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confidence: 67%

On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations

Iandoli

2022

Preprint

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“…Finally, the main assumption in (H3) is satisfied at least in H s (R/ΥZ) for s > 7/2. Indeed, the following theorem is proved in a forthcoming paper [Mie15].…”

Section: Quasilinear Kdvmentioning

confidence: 98%

“…Indeed, the following theorem is proved in a forthcoming paper [Mie15]. In the special case when κ is constant, well-posedness is known to hold true with much lower regularity, for example s > −1/2 is sufficient for the classical KdV [KPV96].…”

Section: This Shows the Equivalence Of Norms Requested In (H2)mentioning

confidence: 99%

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Co-periodic stability of periodic waves in some Hamiltonian PDEs

2016

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The stability theory of periodic traveling waves is much less advanced than for solitary waves, which were first studied by Boussinesq and have received a lot of attention in the last decades. In particular, despite recent breakthroughs regarding periodic waves in reaction-diffusion equations and viscous systems of conservation laws [Johnson-Noble-Rodrigues-Zumbrun, Invent math (2014)], the stability of periodic traveling wave solutions to dispersive PDEs with respect to 'arbitrary' perturbations is still widely open in the absence of a dissipation mechanism. The focus is put here on co-periodic stability of periodic waves, that is, stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Fairly general nonlinearities are allowed in these systems, so as to include various models of mathematical physics, and this precludes complete integrability techniques. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg-de Vries equation (qKdV), and the Euler-Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability, and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. However, the resulting criteria are more complicated for periodic waves because they have more degrees of freedom than solitary waves, so that the action is a function of N + 2 variables for a system of N PDEs, while the moment of instability is a function of the wave speed only once the endstate of the solitary wave is fixed. Regarding solitary waves, the celebrated Grillakis-Shatah-Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, stability criteria involve all the second order, partial derivatives of the action. This had already been pointed out by various authors for some specific equations, in particular the generalized Korteweg-de Vries equation -which is special case of (qKdV) -but not from a general point of view, up to the authors' knowledge. The most striking results obtained here can be summarized as: an odd value for the difference between N and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Furthermore, it is shown that, when applied to the Euler-Korteweg system, this approach yields several interesting connexions between (EKE), (EKL), and (qKdV). More precisely, (EKE) and (EKL) share the same abbreviated a...

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“…More, this seems to be the first instance of a proof of hypocoercive 1 decay where periodicity is used in a crucial way. We note, finally, the relation between these weights and the "gauge functions" used for similar purposes in short-time (i.e., well-posedness) dispersive theory [LP02,BGDD06,Mie15], a connection brought out further by our choice of notation in the proof. This indicates perhaps a potential for wider applications of these ideas in the study of periodic wave trains.…”

Section: Introductionmentioning

confidence: 99%

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

Rodrigues¹,

Zumbrun²

2016

SIAM J. Math. Anal.

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Abstract. A technical obstruction preventing the conclusion of nonlinear stability of large-Froude number roll waves of the St. Venant equations for inclined thin film flow is the "slope condition" of Johnson-Noble-Zumbrun, used to obtain pointwise symmetrizability of the linearized equations and thereby high-frequency resolvent bounds and a crucial H s nonlinear damping estimate. Numerically, this condition is seen to hold for Froude numbers 2 < F 3.5, but to fail for 3.5 F . As hydraulic engineering applications typically involve Froude number 3 F 5, this issue is indeed relevant to practical considerations. Here, we show that the pointwise slope condition can be replaced by an averaged version which holds always, thereby completing the nonlinear theory in the large-F case. The analysis has potentially larger interest as an extension to the periodic case of a type of weighted "Kawashima-type" damping estimate introduced in the asymptotically-constant coefficient case for the study of stability of large-amplitude viscous shock waves.

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

Cited by 5 publications

References 25 publications

On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations

On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations

Co-periodic stability of periodic waves in some Hamiltonian PDEs

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

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