17 research outputs found
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
of the dispersive equation we introduce satisfies local-in-time Strichartz
estimates with loss in derivative:
where depends on and on the norms of the
initial data in . 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 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
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
.Comment: 60 page
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
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
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
Machining of micro aspherical mould inserts
This paper reports a parametric investigation and development of grinding technologies for micro aspherical mould inserts using parallel grinding method. The parametric investigation revealed that at nanometric scale the undeformed chip thickness has little influence on the surface finish of ground inserts. The grinding trace spacing has a slightly larger influence on the surface finish. A new technique was developed to true and dress the resin bonded micro wheels with mesh size of #3000, which produced a satisfactory wheel form accuracy and relatively high grain packing density. A form error compensation technique was also developed, with which mould inserts of submicron form accuracy were consistently produced. Using the developed technologies, micro aspherical inserts of diameters ranging from 200 mu m to 1000 mu m with surface finish of around 10 nm and form error of similar to 0.2-0.4 mu m were successfully fabricated. (c) 2005 Elsevier Inc. All rights reserved
Quadratic Life Span of Periodic Gravity-capillary Water Waves
We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size \u3b5 in a sufficiently smooth Sobolev space leads to a solution that remains in an \u3b5-ball of the same Sobolev space up times of order \u3b5 122. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form