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Long-time existence for multi-dimensional periodic water waves

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Abstract

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 \({\varepsilon}\), smooth solutions exist up to times of the order of \({\varepsilon^{-5/3+}}\), 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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Acknowledgements

We would like to thank J.M. Delort for useful discussions on the topic, and for pointing out the work [DI17] where extended lifespan results are obtained for strongly semilinear KG equations in the presence of small divisors.

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  1. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08540, USA

    A. D. Ionescu

  2. Department of Mathematics, University of Toronto, 40 St George St, Toronto, ON, M5S 2E4, Canada

    F. Pusateri

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Correspondence to F. Pusateri.

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A. D. Ionescu was supported in part by NSF Grant DMS-1600028 and by NSF-FRG Grant DMS-1463753. F. Pusateri was supported in part by Start-up grants from Princeton University and the University of Toronto, and NSERC Grant RGPIN-0648.

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Ionescu, A.D., Pusateri, F. Long-time existence for multi-dimensional periodic water waves. Geom. Funct. Anal. 29, 811–870 (2019). https://doi.org/10.1007/s00039-019-00490-8

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