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

Berti, Massimiliano; Delort, Jean-Marc

doi:10.48550/arxiv.1702.04674

Cited by 9 publications

(57 citation statements)

References 56 publications

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“…In particular, if ω 0 = 0, this gives a proof that the solution can be continued until T ∼ ε −N 0 . See also [20] for a similar lifespan bound for irrotational water waves on a periodic domain. Let us make a few remarks.…”

Section: Introductionmentioning

confidence: 91%

On the lifespan of three-dimensional gravity water waves with vorticity

Ginsberg

2018

Preprint

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We prove a long-term regularity result for three-dimensional gravity water waves with small initial data but nonzero initial vorticity. We consider solutions whose vorticity vanishes on the free boundary and use this to derive a system for the evolution of the free boundary which reduces to the Zakharov/Craig-Sulem formulation in the irrotational case. We are able to continue the solution until a time determined by the size of the initial vorticity in such a way that if the vorticity is zero, one recovers a lifespan T ∼ ε −N where N can be taken arbitrarily large if the initial data is taken to be arbitrarily smooth.

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Section: Introductionmentioning

confidence: 91%

On the lifespan of three-dimensional gravity water waves with vorticity

Ginsberg

2018

Preprint

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“…A new different approach in the case of super-linear dispersion law (i.e. L has order > 1) has been proposed by Berti-Delort in [7] for the capillary water waves equation. The starting point is to rewrite the equation as a para-differential system which involves a para-differential term and a smoothing remainder.…”

Section: Introductionmentioning

confidence: 99%

“…For more details on this strategy we refer the reader to the introduction of [7]. We mention that in [18], [17] it has been shown that a large class of fully non linear Schrödinger type equations admits quasi-periodic in time, and hence globally defined and stable, small amplitude solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Hence it would be interesting to study whether other stability phenomena appear. The goal of this paper is to extend the BNF theory to a class of fully non linear Schrödinger equations by adapting the ideas of [7].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we define the classes of operators and of symbols we need and we develop some composition theorems for classes of para-differential and smoothing operators. Such classes of operators have been introduced and widely studied in [7]. The first step is to rewrite the system (1.15) as a para-differential system of the form (3.1) with U = U (t, x) = (u, ū) by using the results of Section 2.…”

Section: Introductionmentioning

confidence: 99%

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Long time existence for fully nonlinear NLS with small Cauchy data on the circle

Feola¹,

Iandoli²

2021

Annali Scuola Normale Superiore - Classe Di Scienze

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In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that for any initial condition even in x, regular enough and of size ε sufficiently small, the lifespan of the solution is of order ε −N for any N ∈ N if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".The following is a subclass of the previous class made of those operators which are autonomous, i.e. they depend on the variable t only through the function U .Definition 2.5 (Autonomous smoothing operator). We define, according to the notation of Definition 2.2, the class of autonomous non-homogeneous smoothing operator R −ρ K,0,N [r, aut] as the subspace of R −ρ K,0,N [r] made of those maps (U, V ) → R(U )V satisfying estimates (2.8) with K ′ = 0, the time dependence being only through U = U (t). In the same way, we denote by ΣR −ρ K,0,p [r, N, aut] the space of maps (U, V ) → R(U, V ) of the form (2.9) with K ′ = 0 and where the last term belongs to R −ρ K,0,N [r, aut].0,N [r, aut]. This inclusion follows by the multi-linearity of R in each argument, and by estimate (2.6). For further details we refer to the remark after Definition 2.2.3 in [7]. Spaces of MapsIn the following, sometimes, we shall treat operators without having to keep track of the number of lost derivatives in a very precise way. We introduce some further classes.. The following is a subclass of the class defined in 2.16 made of those symbols which depend on the variable t only through the function U .Definition 2.18 (Autonomous non-homogeneous Symbols). We denote by Γ m K,0,p [r, aut] the subspace of Γ m K,0,p [r] made of the non-homogeneous symbols (U, x, ξ) → a(U ; x, ξ) that satisfy estimate (2.18) with K ′ = 0, the time dependence being only through U = U (t).Remark 2.19. A symbol a(U; ·) of Γ m p defines, by restriction to the diagonal, the symbol a(U, . . . , U ; ·) for Γ m K,0,p [r, aut] for any r > 0. For further details we refer the reader to the first remark after Definition 2.1.3 in [7].

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

Abstract: 1 partially supported by PRIN 2012 "Variational and perturbative aspects of nonlinear differential problems".2 partially supported by the ANR project 13-BS01-0010-02 "Analyse asymptotique des équations aux dérivées partielles d'évolution".

Cited by 9 publications

References 56 publications

On the lifespan of three-dimensional gravity water waves with vorticity

On the lifespan of three-dimensional gravity water waves with vorticity

Long time existence for fully nonlinear NLS with small Cauchy data on the circle

Long-time existence for multi-dimensional periodic water waves

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