Strichartz estimates for the water-wave problem with surface tension

Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales depending on the size of the frequencies. We infer that a solution $u$ of the dispersive equation we introduce satisfies local-in-time Strichartz estimates with loss in derivative: $$\| u \|_{L^p([0,T]) W^{s-1/p,q}(\mathbb{R})} \leq C, \qquad \frac{2}{p} + \frac{1}{q} = {1/2},$$ where $C$ depends on $T$ and on the norms of the initial data in $H^s \times H^{s-3/2}$. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.Comment: Fixed typos and mistakes. Merged with arXiv:0809.451

Large time wellposdness to the 3-D Capillary-Gravity Waves in the long wave regime

In the regime of weakly transverse long waves, given long-wave initial data, we prove that the nondimensionalized water wave system in an infinite strip under influence of gravity and surface tension on the upper free interface has a unique solution on [0,{T}/\eps] for some \eps independent of constant $T.$ We shall prove in the subsequent paper \cite{MZZ2} that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev-Petviashvili (KP) equations.Comment: Split the original paper(The long wave approximation to the 3-D capillary-gravity waves) into two parts, this is the first on

Global well-posedness of the 3-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate $1/t$.Comment: 60 page

Large time existence for 3D water-waves and asymptotics

We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves. The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.Comment: Revised version of arXiv:math.AP/0702015 (notations simplified and remarks added) To appear in Inventione

On the finite-time splash and splat singularities for the 3-D free-surface Euler equations

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time "splash" (or "splat") singularity first introduced in [9], wherein the evolving 2-D hypersurface, the moving boundary of the fluid domain, self-intersects at a point (or on surface). Such singularities can occur when the crest of a breaking wave falls unto its trough, or in the study of drop impact upon liquid surfaces. Our approach is founded upon the Lagrangian description of the free-boundary problem, combined with a novel approximation scheme of a finite collection of local coordinate charts; as such we are able to analyze a rather general set of geometries for the evolving 2-D free-surface of the fluid. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems, including compressible flows, plasmas, as well as the inclusion of surface tension effects.Comment: 40 pages, 5 figures, to appear in Comm. Math. Phys, abstract added for UK RE