Quasi-periodic standing wave solutions of gravity-capillary water waves

Berti, Massimiliano; Montalto, Riccardo

doi:10.48550/arxiv.1602.02411

Cited by 28 publications

(62 citation statements)

References 40 publications

(98 reference statements)

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“…Actually the constant coefficient symbols of H are of order 1/2, see the remark after Proposition 4.2.4. This information is not necessary for the subsequent normal form arguments but it is in agreement with the asymptotic expansion of the Floquet exponents of the periodic and quasi-periodic solutions found in [1], [16].…”

Section: Step 2: Reduction To Constant Coefficients At Principal Ordersupporting

confidence: 68%

“…For the problem with surface tension existence of time periodic standing wave solutions has been proved by Alazard-Baldi [1]. Recently Berti-Montalto [16] have extended this result proving the existence of also time quasi-periodic capillarity-gravity standing wave solutions, as well as their linear stability. All the above results prove the existence of solutions when the parameters (g, κ, h) satisfy suitable non-resonance conditions.…”

Section: Theorem 1 (Almost Global Existence Of Periodic Capillarity-g...mentioning

confidence: 88%

“…The construction of a modified energy E s which satisfies (0.4.1) will rely, as already said, on two main conceptually different procedures. First we shall perform a series of non-linear para-differential changes of variables, similar to the transformations used in Alazard-Baldi [1] and Berti-Montalto [16] to reduce the linearized equations (which arise in a Nash-Moser iteration to prove the existence of periodic and quasi-periodic solutions) to constant coefficients. Then we shall develop a normal form method parallel to those used by Bambusi, Delort, Grébert and Szeftel [13], [14], [30], and Faou-Grébert [35] for semi-linear PDEs.…”

Section: Reduction To Constant Coefficientsmentioning

confidence: 99%

“…even) function of ξ, so has to vanish. This remark is not necessary for the subsequent arguments but it provides the expected asymptotic expansion of the Floquet exponents in the periodic and quasiperiodic case [1], [16].…”

Section: ) Parity Preserving (316) and Anti-reversibility (318) Prope...mentioning

confidence: 99%

“…This step is essential, otherwise the transformations performed above to decrease the size of the nonlinearity, like u + Q(u, ū), would be unbounded. We perform such a reduction along the lines of the recent works of Alazard-Baldi [1] and Berti-Montalto [16] concerning respectively periodic and quasiperiodic solutions of (0.1.1), ending up with a system as…”

Section: Introduction To the Proofmentioning

confidence: 99%

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Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions

Berti,

Delort

2017

Preprint

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Section: Step 2: Reduction To Constant Coefficients At Principal Ordersupporting

confidence: 68%

Section: Theorem 1 (Almost Global Existence Of Periodic Capillarity-g...mentioning

confidence: 88%

Section: Reduction To Constant Coefficientsmentioning

confidence: 99%

Section: ) Parity Preserving (316) and Anti-reversibility (318) Prope...mentioning

confidence: 99%

Section: Introduction To the Proofmentioning

confidence: 99%

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Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions

Berti,

Delort

2017

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Quasi-periodic solutions of forced Kirchhoff equation

Montalto

2017

Nonlinear Differ. Equ. Appl.

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In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchoff equation with periodic boundary conditions. This is the first KAM result for a quasi-linear wave-type equation. The main difficulties are: (i) the presence of the highest order derivative in the nonlinearity which does not allow to apply the classical KAM scheme, (ii) the presence of double resonances, due to the double multiplicity of the eigenvalues of −∂ xx . The proof is based on a Nash-Moser scheme in Sobolev class. The main point concerns the invertibility of the linearized operator at any approximate solution and the proof of tame estimates for its inverse in high Sobolev norm. To this aim, we conjugate the linearized operator to a 2 × 2, time independent, block-diagonal operator. This is achieved by using changes of variables induced by diffeomorphisms of the torus, pseudo-differential operators and a KAM reducibility scheme in Sobolev class.We consider the Kirchoff equation in 1-dimension with periodic boundary conditionsOur aim is to prove the existence and the linear stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for δ small enough and for ω in a suitable Cantor like set of parameters with asymptotically full Lebesgue measure.The Kirchoff equation has been introduced for the first time in 1876 by Kirchoff, in dimension 1, without forcing term and with Dirichlet boundary conditions, namelyto describe the transversal free vibrations of a clamped string in which the dependence of the tension on the deformation cannot be neglected. It is a quasi-linear PDE, namely the nonlinear part of the equation contains as many derivatives as the linear differential operator. The Cauchy problem for the Kirchoff equation (also in higher dimension) has been extensively studied, starting from the pioneering paper of Bernstein [9]. Both local and global existence results have been established for initial data in Sobolev and analytic class, see [2], [3], [23], [24], [32], [40], [41]. Concernig the existence of periodic solutions, Kirchoff himself observed that the equation (1.2) admits a sequence of normal modes, namely solutions of the form v(t, xUnder the presence of the forcing term f (t, x) the normal modes do not persist, since, expanding v(t, x) = j v j (t) sin(jx), f (t, x) = j f j (t) sin(jx), all the components v j (t) are coupled in the integral term T |∂ x v| 2 dx and the equation (1.2) is equivalent to the infinitely many coupled ODEs v ′′ j (t) + j 2 v j (t) 1 + k k 2 |v k (t)| 2 = f j (t) , j = 1, 2, . . . .The existence of periodic solutions for the Kirchoff equation, also in higher dimension, have been proved by Baldi in [4], both for Dirichlet boundary conditions (v = 0 on ∂Ω) and for periodic boundary conditions (Ω = T d ). This result is proven via Nash-Moser method and thanks to the special structure of the nonlinearity (it is diagonal in space), the linearized operator at any approximate solution can be inverted ...

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Long-time existence for multi-dimensional periodic water waves

Ionescu

Pusateri

2019

Geom. Funct. Anal.

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We prove an extended lifespan result for the full gravity-capillary water waves system with a 2 dimensional periodic interface: for initial data of sufficiently small size ε, smooth solutions exist up to times of the order of ε −5/3+ , for almost all values of the gravity and surface tension parameters.Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of the equations which gives additional smoothing close to the resonant hypersurfaces, (2) another structural property, connected to time-reversibility, that allows us to handle "trivial" cubic resonances, (3) sharp small divisors lower bounds on three and four-way modulation functions based on counting arguments, and (4) partial normal form transformations and symmetrization arguments in the Fourier space. Our theorem appears to be the first extended lifespan result for quasilinear equations with non-trivial resonances on a multi-dimensional torus.

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Quasi-periodic standing wave solutions of gravity-capillary water waves

Cited by 28 publications

References 40 publications

Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions

Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions

Quasi-periodic solutions of forced Kirchhoff equation

Long-time existence for multi-dimensional periodic water waves

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