Mei Ming, Zhifei Zhang
In this paper, we prove the local well-posedness of the water wave problem with surface tension in the case of finite depth by working in the Eulerian setting. For the flat bottom, as surface tension tends to zero, the solution of the water wave problem with surface tension converges to the solution of the water wave problem without surface tension.
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This is the second paper in a series of papers analyzing angled crested type water waves with surface tension. We consider the 2D capillary gravity water wave equation and assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the first paper \cite{Ag19} we constructed a weighted energy which generalizes the energy of Kinsey and Wu \cite{KiWu18} to the case of non-zero surface tension, and proved a local wellposedness result. In this paper we prove that under a suitable scaling regime, the zero surface tension limit of these solutions with surface tensi…
This paper has been merged to 0908.3255.
Based on the a priori estimates in our previous work \cite{MW}, we continue to investigate the water-waves problem in a bounded two dimensional corner domain in this paper. We prove the local well-posedness of the solution to the water-waves system when the contact angles are less than $\fracπ{16}$.
Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates.
Regularizing effects of surface tension are studied for interfacial waves between a two-dimensional, infinitely-deep and irrotational flow of water and vacuum. The water wave problem under the influence of surface tension is formulated as a system of second-order in time nonlinear dispersive equations. The main result states that if the curvature of the initial fluid surface is in the Sobolev space of order k+7/2 and if its derivatives decay faster than a polynomial of degree k+1 does then the curvature of the fluid surface corresponding to the solution of the problem instantaneously gains k/2…
We consider the capillary-gravity water wave equation in two dimensions. We assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. We construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. When the surface tension $σ$ is zero, the energy reduces to a lower order version of the energy obtained by Kinsey and Wu [23] and allows angled crest interfaces. For positive surface tension, the energy does not allow angled crest interfaces but admits initial data with large curvature of the order of $σ^{-\frac…
Local Well-Posedness of the Inviscid Shallow Water Equations with Surface Tension
该研究证明带重力和表面张力的无粘浅水方程在一、二维空间局部适定，通过利用系统对称性在实值相空间给出替代证明，其成果可用于数值格式设计等后续研究。
We consider the two-dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. In order to describe the evolution of the envelopes of small oscillating wave packet-like solutions to this problem the Nonlinear Schrödinger equation can be derived as a formal approximation equation. In recent years, the validity of this approximation has been proven by several authors for the case without surface tension. In this paper, we rigorously justify the Nonlinear Schrödinger approximation for the cases with and without surface tension by proving erro…
We study the modulational instability of a shallow water model, with and without surface tension, which generalizes the Whitham equation to include bi-directional propagation. Without surface tension, the small amplitude periodic traveling waves are modulationally unstable if their wave number is greater than a critical wave number predicting a Benjamin-Feir type instability and the result qualitatively agrees with the shallow water model in [HP16]. With surface tension, the result qualitatively agrees with the physical problem except for the large surface tension limit which is accurately pre…
We consider the motion of a two-dimensional interface between air (above) and an irrotational, incompressible, inviscid, infinitely deep water (below), with surface tension present. We propose a new way to reduce the original problem into an equivalent quasilinear system which are related to the interface's tangent angle and a quantity related to the difference of tangential velocities of the interface in the Lagrangian and the arc-length coordinates. The new way is relatively simple because it involves only taking differentiation and the real and the imaginary parts. Then if assuming that wav…
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^{…
The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. Métivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrödinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V…
The traveling wave problem for the shallow water equations: well-posedness and the limits of vanishing viscosity and surface tension
本文研究带阻尼浅水系统的孤立行波解，发展了小数据适定性理论，并利用 Nash-Moser 隐函数定理证明行波解在有无粘性或表面张力及所有非平凡速度下均持续存在，且在消失极限下保持强范数连续性。
We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method, based on energy estimates and dispersive analysis, which allows us to deal with strong singularities arising from time resonances in the applications of the normal form method and nonlinear scattering. As a result, we are able to consider a suitable class of pertur…
The purpose of this paper is to derive rigorously the so called viscous shallow water equations given for instance page 958-959 in [A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of equations is similar to compressible Navier-Stokes equations for a barotropic fluid with a non-constant viscosity. To do that, we consider a layer of incompressible and Newtonian fluid which is relatively thin, assuming no surface tension at the free surface. The motion of the fluid is described by 3d Navier-Stokes equations with constant viscosity and free surface. We prove t…
This paper is devoted to the stabilization of the water-wave equations with surface tension through of an external pressure acting on a small part of the free surface. It is proved that the energy decays to zero exponentially in time, provided that the external pressure is given by the normal component of the velocity at the free surface multiplied by an appropriate cut-off function.
In this paper, we mainly consider two class of travelling stratified periodic water waves, one with negative (or without) surface tension and the other with constant Bernoulli's function and stagnation points. We first establish the symmetry result for stratified water waves with negative (or without) surface tension, but without stagnation by using the modified maximum principle. Furthermore, the symmetry property of stratified water waves with constant Bernoulli's function and stagnation points is also obtained provided the monotonic property is known.
Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits
本文研究了具有涡旋的可压缩重力-表面张力水波，通过精心设计的近似系统和双曲方法证明了局部well-posed性，同时获得不可压缩和零表面张力极限。
We present an exact relation among the kinetic, potential and surface tension energies of a solitary wave in deep water in all dimensions. We deduce its non-existence in the absence of the effects of surface tension, provided that gravity acts in a direction opposite to what is physically realistic.
We study the two dimensional water waves problem with surface tension in the case when there is a non-zero contact angle between the free surface and the bottom. In the presence of surface tension, dissipations take place at the contact point. Moreover, when the contact angle is less than $π/6$, no singularity appears in our settings. Using elliptic estimates in corner domains and a geometric approach, we prove an a priori estimate for the water waves problem.
We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.
A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Preci…
Fully localised three-dimensional solitary water waves on Beltrami flows with strong surface tension
本文研究了在Beltrami流中强表面张力条件下，三维孤立水波的局域化存在性及KP-I方程的扰动解
The main result of this work is the construction of multi-solitons solutions that is to say solutions that are time asymptotics to a sum of decoupling solitary waves for the full water waves system with surface tension.
Some effects of surface tension on fully-nonlinear, long, surface water waves are studied by numerical means. The differences between various solitary waves and their interactions in subcritical and supercritical surface tension regimes are presented. Analytical expressions for new peaked travelling wave solutions are presented in the case of critical surface tension. The numerical experiments were performed using a high-accurate finite element method based on smooth cubic splines and the four-stage, classical, explicit Runge-Kutta method of order four.
We prove the local existence for the Water Waves equations with large bathymetric variations on a time interval of size 1/ε, where $ε$ measures the amplitude of the wave. We just need the presence of surface tension.
In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.
This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the s…
Local well-posedness and global stability of one-dimensional shallow water equations with surface tension and constant contact angle
本文研究具有毛细表面和移动接触线的一维浅水问题，从二维水波方程导出能量模型，并针对自由边界处的退化与奇点建立了平衡态的全局稳定性及局部适定性理论。
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 3-waves 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 $ε$ in a sufficiently smooth Sobolev space lead to a solution that remain…
This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.
This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.
We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability of gravity waves. In the case of weak surface tension, we find inter…
Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator i…
Stratified equatorial flows in cylindrical coordinates with surface tension
本文研究了柱坐标下具有自由表面的无粘不可压缩流体在赤道区域的稳定流动，考虑了重力和表面张力作用下的波传播，并证明了自由表面的正则性结果。
We prove the first bifurcation result of time quasi-periodic traveling waves solutions for space periodic water waves with vorticity. In particular we prove existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restricting the surface tension to a Borel set of asymptotically full Lebesgue measure.
It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength t…
In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in Hölder spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.
Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $β$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<β<\frac{1}{3}$. A classical vari…
On the free-boundary Incompressible Elastodynamics with and without surface tension
本研究针对含表面张力的自由边界不可压缩弹性动力学问题，拓展已有不可压缩磁流体动力学研究证明拉格朗日坐标下局部适定性，借鉴可压缩重力-毛细水波思路得到图形坐标下一致能量估计，推导了零表面张力极限。
We prove the existence of small steady periodic capillary-gravity water waves for general stratified flows, where we allow for stagnation points in the flow. We establish the existence of both laminar and non-laminar flow solutions for the governing equations. This is achieved by using bifurcation theory and estimates based on the ellipticity of the system, where we regard, in turn, the mass-flux and surface tension as bifurcation parameters.
In this paper, we introduce a new extended version of the shallow water equations with surface tension which is skew-symmetric with respect to the L2 scalar product and allows for large gradients of fluid height. This result is a generalization of the results published by P. Noble and J.-P. Vila in [SIAM J. Num. Anal. (2016)] and by D. Bresch, F. Couderc, P. Noble and J.P. Vila in [C.R. Acad. Sciences Paris (2016)] which are restricted to quadratic forms of the capillary energy respectively in the one dimensional and two dimensional setting.This is also an improvement of the results by J. Lall…
We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces.
We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The proof of well-posedness relies on energy estimates. However, due to the symmetry lack of the nonlinear part, in order to close the a priori estimates one has to modify the traditional energy norm in use. Hamiltonian conservation provides with global well-posedness at least for small initial data in the one dimensional settings.
In this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension two (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time-resonance structure and modified scattering. It is slightly simpler, however, at the technical level and our goal here is to present our method in this simplified situation. The full system is the subject of a forthcoming paper.
We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+ε}$-class for some $ε>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.
We consider the capillary-gravity water-waves problem of finite depth with a flat bottom of one or two horizontal dimensions. We derive the modulation equations of leading and next-to-leading order in the hyperbolic scaling for three weakly amplitude-modulated plane-wave solutions of the linearized problem in the absence of quadratic and cubic resonances. We fully justify the derived system of macroscopic equations in the case of pure gravity waves, i.e. in the case of zero surface tension, employing the stability of the water-waves problem on the time-scale $O(1/\ep)$ obtained by Alvarez-Sama…
On the Cauchy Problem of Spherical Capillary Water Waves
本文通过紧李群上的仿微分演算，将球面毛细水波方程对称化为1.5阶拟线性色散方程，从而在更弱正则性条件下证明了其局部适定性。
We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - μ$, where $μ= \frac{1}{10}$ in the case $d = 1$ and $μ= \frac{1}{5}$ in the case $d \geq 2$, extending previous results of Alazard-Burq-Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.
Control of three dimensional water waves
本文研究三维空间周期水波在表面张力下的精确可控性，通过局部外部压力控制，证明在几何控制条件下任意短时间内可控，方法结合迭代方案与半经典方法。
We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth.
From KP-I Lump Solution to Travelling wave of 3D Gravity Capillary Water wave problem
本文通过KP-I方程模型，证明三维重力毛细水波问题在特定参数下存在类似团块解的完全局域孤立行波。
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 p…
We prove local exact controllability in arbitrary short time of the two-dimensional incompressible Euler equation with free surface, in the case with surface tension. This proves that one can generate arbitrary small amplitude periodic gravity-capillary water waves by blowing on a localized portion of the free surface of a liquid.
We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each c…
This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is vacuum, and that the waves are acted upon by gravity with surface tension effects on the air--water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted $δ$-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of $0$. Our mai…
We consider small amplitude wave packet-like solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Formal multiscale calculations suggest that the modulation of such a solution is described by a profile traveling at group velocity and governed by a hyperbolic cubic nonlinear Schrödinger equation. In this paper we show that, given wave packet initial data, the corresponding solution exists and retains the form of a wave packet on natural NLS time scales. Moreover, we give rigorous error estimates between the true and formal soluti…
We consider the free boundary problem for a layer of viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom and below the atmosphere. For the "semi-small" initial data, we prove the zero surface tension limit of the problem within a local time interval. The unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization. For the small initial data, we prove the global-in-time zero surface tension limit of the problem.
Starting from the paper by Dias, Dyachenko and Zakharov (\emph{Physics Letters A, 2008}) on viscous water waves, we derive a model that describes water waves with viscosity moving in deep water with or without surface tension effects. This equation takes the form of a nonlocal fourth order wave equation and retains the main contributions to the dynamics of the free surface. Then, we prove the well-posedness in Sobolev spaces of such equation.
In this paper we derive some new weakly nonlinear asymptotic models describing viscous waves in deep water with or without surface tension effects. These asymptotic models take into account several different dissipative effects and are obtained from the free boundary problems formulated in the works of Dias, Dyachenko and Zakharov (Physics Letters A, 2008), Jiang, Ting, Perlin and Schultz (Journal of Fluid Mechanics,1996) and Wu, Liu and Yue (Journal of Fluid Mechanics, 2006).
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