Journal
GEOMETRIC AND FUNCTIONAL ANALYSIS
Volume 29, Issue 3, Pages 811-870Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00039-019-00490-8
Keywords
-
Categories
Funding
- NSF [DMS-1600028]
- NSF-FRG Grant [DMS-1463753]
- Princeton University
- University of Toronto
- NSERC [RGPIN-0648]
Ask authors/readers for more resources
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 epsilon, smooth solutions exist up to times of the order of epsilon-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.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available