Gravity capillary standing water waves

Abstract

The paper deals with the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the question of the nonlinear interaction of a plane wave with its reflection off a vertical wall. The main result is the construction of small amplitude, standing (namely periodic in time and space, and not travelling) solutions of Sobolev regularity, for almost all values of the surface tension coefficient, and for a large set of time-frequencies. This is an existence result for a quasi-linear, Hamiltonian, reversible system of two autonomous pseudo-PDEs with small divisors. The proof is a combination of different techniques, such as a Nash-Moser scheme, microlocal analysis, and bifurcation analysis.

80 pages

References in corpus (6)

• On the Water Waves Equations with Surface Tension
• KAM for autonomous quasi-linear perturbations of mKdV
• KAM for reversible derivative wave equations
• Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves
• Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type
• Multi-solitons and related solutions for the water-waves system

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• Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
• Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions
• Quasi-periodic solutions for the forced Kirchhoff equation on ${\mathbb{T}}^d$
• Long time existence for fully nonlinear NLS with small Cauchy data on the circle
• A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension
• A priori estimates for the free-boundary Euler equations with surface tension in three dimensions
• Harmonic Stability of Standing Water Waves
• A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid
• Explicit Dirichlet-neumann Operator For Water Waves
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• Quasi-periodic solutions of forced Kirchoff equation
• Nonexistence of small doubly periodic solutions for dispersive equations
• Paralinearization of free boundary problems in fluid dynamics
• The semi-analytic theory and computation of finite-depth standing water waves
• Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations
• Regularity results on the flow maps of periodic dispersive Burgers type equations and the Gravity-Capillary equations